For the technical bit: You need streaming computers (aka servers, physical or virtual doesn't matter) which get fed by your main computer with a clean feed (no overlays) and feed the streaming platforms you want to stream to with the individual delay settings and overlays you want to.
For users with a small to near no fan base, this makes gaining maximum traction on a stream more difficult than it needs to be.
While an exclusivity clause is annoying, it's not unique to facebook: Twitch has the very same policy for their partners, YouTube does not.
If you interact with your chat at all, you generally want to avoid to stream on multiple platforms at once, and you especially don't want to stream on multiple platforms with different delay settings.
Unlike other forms of content, the strength of live content with viewer interaction is the building of communities: You may just have 5 viewers regularly appearing in chat, but you can count on them appearing and chatting with each other and you each time you stream. If you have the same 5 viewers split over 3 platforms for the sake of discoverability, you won't get any chatting between them, and it's way more likely that random people stopping by will just leave again because just-you isn't interesting enough for them whereas you+chat might be. Different delays make it even more confusing to what people in chat are referring to than it already is.
If you don't interact with chat at all, you need to carry the show on your own, which is a lot harder – effectively, you're doing a VOD but without the option to cut out the boring bits. I don't recommend going this way.
So, to grow your fanbase, you don't really need (or rather: you really don't need) to stream on every platform at once, but instead focus on building a community in which as many people as possible feel welcome.
Alas, we've moved well beyond the scope of this forum here, so I'm going to just drop some links for further reading and discussion