Wow, really touchy issue.
Since you said you already have a pretty much nonexistent relationship with this person, can I assume it's not just a case of drifting apart for no reason? Can we basically assume there is a reason you are estranged from them? If so, that alone tells me you should probably simply send them whatever amount of cash you are comfortable sending, with a note saying you will help out this time (especially if it's true as some folks said that you have plenty of cash so it won't be a burden on you), but that they should really not ask for any more money because you'll just not respond.
Then inform them it's not a "loan", that you are giving them this money to help out because they need it, they are family, and you can afford it at the moment. If they still chose to send you money to repay the debt, great. If not, though, you've already assumed it won't be paid back. I agree strongly with Majic Walrus on this, just assume it won't be paid back. And tell them you don't want it paid back.
I've had family members ask me for money. Sometimes it's someone I totally trust and I don't even think about loaning or giving them the money. Usually, it's for a temporary thing like not having enough to pay a dinner bill or something immediate like that, where I get paid back within a day or so. Other times I've had a sibling ask for a loan for bills, and I made the loan to one of them but not to others. The ones I refused were due to having a very strained relationship and suspecting they wouldn't pay bills but would instead buy drugs with it. I recommend NEVER giving money to family or friends if you have any suspicion that the money is going to anything other than what you're told it's for.
If you know for sure it's really a need for money to pay bills, is it a situation that will improve in the future, or will it just recur as a problem for them again? Will they make good decisions about how best to use the money in their bill situation? And do they have kids who will be harmed if they don't get the financial assistance? If they really need it for bills and will apply the money properly, and if there are also kids involved, then that's important. Is the cause of your estrangement something of a serious nature -- abuse, for example -- or is it just a more typical family estrangement issue? If it's the latter, then it doesn't mean they are a bad person or someone you don't care about at all, right? You can love someone in your family even if you don't like them. So wanting to help a person truly in need (and especially if kids are involved) can outweigh personal dislike.
If the sum is a large amount, and you are willing to provide it, making it a gift instead of a loan has the added benefit of not creating additional tension in an already strained relationship. Even among friends and family you are on good terms with, a sizeable "loan" that is expected to be paid back can make things weird if they are unable to pay it back as soon as expected, and there's the awkwardness of whether they mention it or you do, it can cause resentments (even if you don't want it to, it's sometimes hard to help feeling that way, and it could be them who feel resentment over feeling embarrassed or guilty etc, it might be irrational but humans are humans and we feel irrational things sometimes).
It's worth also remembering, though, that your fiancee should actually have some say in the matter -- you're going to be married, and financial decisions (regardless of who makes the most money, keeping separate finances, whatever) should be joint decisions even when or especially when it's family and friends. It can cause tensions and resentments between you and your partner, the repercussions (family getting upset or resentful, asking for more money, etc) will affect your fiancee as much as you, and you should ask yourself if doing this for your family member is worth the potential disagreements and potential fallout that could unduly affect your relationship with your fiancee. People tend to think "it's my money, it's my family" but you're making a life with this other person and how your decision affects the two of you is important.
If you give the money, you might want to talk to your fiancee and ask what would make them feel comfortable with the loan. Ask them to set parameters for the situation, what sort of amount they are most comfortable with, and assure them you want to make the decision in a way that you are both able to accept. Tell them you will make it clear to your family that this is absolutely a one-time thing, and that if another request for money does come then you will refuse it.
I suspect part of your fiancee's reluctance is due to the nature of your relationship with this family member, mixed with some resentment that the family member would ask you for money despite your strained relationship. And your fiancee also likely sees this as a sort of test-run for how things could play out when you are married if you are caught between a family request and your fiancee's disagreement with it. Right or wrong, it probably appears to your fiancee to be a question of whether your strained relationship with a family member is still going to outweigh your good relationship with your fiancee. Competing wishes.
It'd be much easier if there are kids involved and if the estrangement isn't due to anything that makes them a really bad person. Your fiancee would probably be much easier to convince to come to some sort of compromise if they understand the kid aspect and that your dislike for this relative isn't due to a really big issue that goes to the relative's character. And, if it is a more typical family estrangement problem (they acted like a jerk a lot, they were insensitive, they make a lot of bad decisions and expect other family to bail them out, etc) then there is the chance that this could, if you want it to, be the start of overcoming some of that estrangement to at least have a more normal, distant but not poisoned relationship with the relative. That would be another big motivating factor that could really be important to explain to your finacee, too.
It's a pain, but it's very common and can be frustrating if it's not handled delicately and in a way clearly demonstrating to your fiancee that you value their wishes and want to find a way to make them happy with the decision. That is in fact probably ultimately where the real issue lies anyway between you and your fiancee. Help them understand, avoid making it an argument, and demonstrate that you really do want them to be comfortable with the decision, that's my best advice. Sorry it's so long.
Best of luck!
