Well, first, let's look at what the champion has to provide to make use of the advantages a team looks for from the Bottom Lane Carry position, being:
- Consistent Sustained Damage (hard for melee as that means consistently getting into range to deal damage and staying there and not running out of damage after one rotation or even two unless they have really low cool downs)
- Objective Control (More or less can kill objectives fast)
- Can safely farm in lane without relying on resources (will they run out of mana farming?) or sustain through the poke (and lots of poke at that)
- Optionally, can synergize a TON with a Support present with them
(these overall line up with your needs)
So, from my experience, that would be:

can safely farm from a distance using his Q due to it's low cool down and can sustain through most poke that comes his way since he can itemize entirely into defenses early on while still dealing a ton of damage. His sustained damage is actually quite exceptional as long as you can stick to your enemies, which he can pretty well due to the 30% Tenacity given to him by his W, the slow from his Q, and the movement speed granted to him by his R. His objective control is VERY strong with monsters due to his Q, but becomes a little more questionable when it comes to structures although the Demolish rune and his E's bonus AD can help with this issue somewhat. He will still be weaker than an ADC, but his increased durability means he does not share their common weakness against burst damage, which he is notorious for shrugging off. Additionally, his signature item Spirit Visage synergizes heavily with supports like Sona and Soraka. It is a rather extreme choice, but I feel could be surprisingly effective as long as you and your team ban any hard counters, like Vayne, Fiora, and any GW appliers.
Note: I would not run him if your team is running a Tank in the Top Lane unless you just have some absolutely insane Jungle DPS champion. That's just far too much potential for someone on the enemy team to pick a % health champion and shred the main advantage that team would have.
's Q allows him to farm extremely quickly from a decent distance and, when combined with the Kleptomancy keystone, also givers him good poke without sacrificing too much farm. FURTHERMORE, his W grants him both the sustain and the reliability to get keep dealing sustained damage that many melee characters would normally suffer from against a ranged match-up. His barrels give him great late game damage and his ultimate allows for great map control, which helps with his main downside in comparison to an ADC, being Objective Control. He is MUCH slower than an ADC in killing Dragon and especially in killing Towers, but his R's map control and his Passive's true damage somewhat help counteract this weakness, although it's still a far cry from an ADC. He doesn't really have any huge Support synergy, but his Q can proc the bonus damage granted by the likes of Sona, Nami, and Janna.
Probably the safest choice here to pick in my opinion, as his counters are limited compared to others here.
I don't feel like this needs explanation, and you said you won't be doing this.
as you said DOES have potential here, but I feel is a bit of a risky pick as, like GP but to the more extreme, he has NO objective control unless you count using your R on monsters as Objective Control. He can't push towers for crap and he is no better than anyone else at quickly killing an Epic Monster. However, his Q, if in the Danger Zone and if Rumble is allowed to farm items, is outstanding. You can't ignore 473 (+165% AP) magic damage over 3 seconds. His W also gives him some sustain against poke (as well as movement speed to stick, although I wouldn't use it for solely movement speed unless you CAN) and his E helps with farming and sticking to enemies (and the damage is decent too). His R is tricky to use, but works extremely well with other Supports with heavy crowd control, such as Blitzcrank, Leona, Nautilus, and Thresh. Again, his Objective Control is absolute hot garbage when it comes to the speed he kills them, but his other advantages warrant his appearance in this list.
is by far the most risky pick of this list as he CAN potentially run OOM in lane if he is bullied a ton since he will be relying a ton on his W for farming if he is getting bullied. He is situational and should NOT be used for a OTP Melee Bot. HOWEVER, he undeniably has the best Objective Control of anyone present on this list and is pretty reliable to stick to someone as long as he can get into his E range. He is a very situational pick, but I would keep him in my back pocket to really abuse a team that won't be able to abuse his weaknesses in this position, which you can probably guess: large amounts of crowd control/peel and early game ADC bullies.
Just wanted to remind you that he is 100% an option, although if you don't wish to use him, don't.
I do not include
as they kind of cheat and aren't full melee, but they can probably be run in Bot.
Champions that are are traps to consider bringing Bottom Lane in my opinion are:
Unless your friend is a Zilean OTP and you only run Duo Queue, I really wouldn't recommend doing this as he just gets bullied SO HARD by a large cast of the ADC roster. He also has VERY little killing power on Dragon and Towers, so running him slows how much reward you can get after a won team fight by a lot. You might have a few good games, but the large amount will likely end poorly.
Fiora's main strength is her ability to abuse someone in lane that she can easily dash around, and also in a DUEL where outside interference is limited. You might see her W as a strong potential answer to poke, but it's CD is too long for the constant poke of Bottom Lane and they will eventually learn to just not use CC during your W since most CC in Bottom Lane is from a range, both removing the pressure of needing to CC Fiora since she is right on top of you (like in Top) and ALSO making it less likely to hit them with the Parry since they are further away and have more time to dodge it.
Most of Irelia's damage is single target and her ability to farm safely is nonexistent. See Kled and Pantheon. She just doesn't work, trust me.
This MIGHT be able to work, but his lack of ability to safely farm from a range makes me doubt this. Jax is likely better just farming up Top where someone isn't constantly basic attacking him from far away. And yes, I know his E stops it, but like Fiora, it just isn't up enough early game for it to justify him being here.
Simply not enough consistent farm to put him here. If you're only plan is to kill the enemy, maybe, but otherwise just don't. Not to mention, unlike in Top Lane, when Kled gets knocked off Skarl, you are basically forced to go back to base as the enemy isn't going to be letting you basic attack them or be in range to hit a ton of Q shots.
See Kled. Only goal is to kill? Sure. However, he suffers in every single other department that you would bring an ADC into Bottom Lane for. He will eventually run OOM if he uses Q to farm, so that's not an option.
Will get poked out of lane and relies too much on quickly getting to Level 6 to put him into a shared XP lane.
HAHAHAHAHAHAHHAHAHAHAHHAHAHAHAHAHA! Funny joke there. His Wither may cripple an ADC late game, but good luck getting many stacks at all, which is one of the main reasons you play this guy. Also, his W doesn't really just destroy an ADC early game unless you max it which... good luck trying W max first as Nasus. Hey, maybe if the enemy is doing the same thing and it's two Melee Bots, maybe try this! Otherwise, no, I really wouldn't recommend it. You can try a weird AP Nasus, but at that point why aren't you just trying a Mage Bottom Lane instead.
Some champions I'm not sure on, so I'm not going in detail on them but recommend investigating:
