So if it came out of an RV, it is most likely not marine rated, and therefore not spark protected. If you don't want to blow yourself up, do not put it on your boat! (assuming you have gasoline engines, if you have diesel, possibly different answer.)
To actually answer your question, the converter is really just a glorified battery charger and should run direct to the battery (with a fuse). Your marine rated inverter should also be wired direct to the battery, with an appropriate fuse. They don't have any interaction and shouldn't be wired to each other. If you want outlets to run on either source, you need a transfer switch which lets you pick a primary and backup 120v source for your entire 120v system.
(there are some RV converters with transfer switches built in, some converters also have inverters with transfer built in, and some stand alone inverters have 120v transfers built in... All vastly different wiring scenarios, but without more info, can't say if any of those apply to you.)
And lastly, if it came from an older RV and weighed a ton, it is best served in the recycle bin. If it is an old MagneTek or B-W mfg converter, those things barely charged (like 3 amps) and would cook a battery like an egg, as they age their voltage would creep up. Mine was passing 16v when I tore it out of my 1985 RV last fall. Newer converters are much lighter and have digital multi-stage chargers.
