I just received a somewhat large vassal after defeating them in a war with my alliance. Now when I check the subject integration it says it costs 1105 influence but the absolute maximum I can ever reach is 1k ... is there a way to lower the integration cost or will I never be able to integrate them (in a non game modifying way)?
There is bit of misunderstanding in your sentence:
Integration it says it costs 1105 influence but the absolute maximum I can ever reach is 1k
The integration cost can go well over 1k and indeed the maximum accumulated influence limit is just 1000, but you pay for integration on monthly basis -5 points per month until you'll pay it all, not a total number all at once. So it is not as hard to offset that cost by other means like declaring rivalries, gaining support from factions, completing your ruler's mandate etc.
I was going to write here about the Vassal Acculturation, but according to wiki it has been changed - instead of reducing the integration cost by 33% now it provides Unity bonus: