That is a problem, but I imagine that without constantly demanding men and resourses for war, they would be able to devote those same men and resources to converting the populace. Being a state based around religious intolerance (realistically on both sides), I imagine that as soon as their situation no longer demanded that they tolerate other religions in their lands, they wouldn't, so conversion (forced or otherwise) would be near the top of the govenment to do list
Except that the situation will always demand that, and devoting the men and resources to converting the populace is the same as internal war.
People are not going to choose conversion over opposition if opposition is viable, and making that untrue would require a far stronger CS than they could possibly have been after Hattin and Muslim states gimped to the point of irrelevance - and if the Muslims and native Christians just leave instead, that's a serious blow to the population of the CS.
I suppose I fail to see much differance between the two situations, since the Europeans were not too keen on travelling to north america either. The difference IMO is that the new settlers were not immediately eaten alive by the war effort to combat the natives, since the North American natives were nothing compared to the Islamic nations. The collonies were either founded by opportunists looking to make money, or religious zealots (and some who were both), which is definitely allot like the CS, since they had the Italians (money and opportunism) and the French/English (religious zealots)
Except that they were a lot more keen on doing so than going to the Levant. Except that North America didn't have a large native population unfriendly to the Europeans and on the same tech level as them and able to fight them on their terms.
Firstly, I dont mean help as in send armies in to give land back to the empire, I mean help by being a stable and non-expansionist country on the frontier. Secondly, of course they will stop supporting the CS whenever it suits them, but that may or may not be soon, and if it isn't soon, then the CS may have time to develope it's own infrastructure and possibly centralize somewhat.
So, utterly unlike OTL, in all regards.