I don't think you should worry about this. You should be fine.
First off, dodging will always take away 3 LP (more for consecutive dodges within a short time), and if this would happen to bring you below 0 LP, then it will. Riot does this so that there is still an LP penalty for those who dodge at 0 LP. Essentially this is means that for that player's next win, they would gain the LP they win minus the LP they lost for dodging while at 0. I'm 99% sure this is intended. Also, you can't go negative in LP from losing either, you just stop at 0 (or below 0 if you are negative from dodging).
Secondly, losing a match while at 0 LP isn't always a guaranteed demotion. Usually it takes multiple lost games at 0 LP to get demoted, but it depends on what your MMR is. Also, in your case, being demoted from Silver V to Bronze I (or being demoted from any V division to a I division) is a lot harder to have happen than a normal demotion. For you to be demoted from Silver V to Bronze I, your MMR needs to be around the average MMR for Bronze V, which basically means that you need to lose a lot of games before being demoted. Pretty much it's incredibly difficult to actually be demoted tiers unless you're actually trying to.