first to say I have a lot of inner conflicts to resolve and I'm working on it, advice on it also greatly appreciated, but now on the situation.

I've moved to another city/country and broke it off with my ltr. I had finally gotten used to the high stress environment job and wanted to get some rotation going. I have mood swings, especially when some things are out of order, for example not hitting the gym often, job change, sickness etc. so there was a period for about a month where I was satisfied with my life. I had gotten a plate and 3 potential ones.

after I've changed jobs I kind of fell of the track and broke it off with them. I managed to keep one going with a very busy girl so the sex was fairly inconsistent. so I was away a couple of days and I came home exhausted and hit her up. she was mentioning going together to the gym for quite some time. ok, have actually no problems with her being a fwb, but considering our busy schedules the sex would be even more inconsistent. anyways, I said we can go the gym just for the sake of it. I know I've fucked up because I didn't think it through and because she's my only source of sex (working on it). so we're done with the gym, it's pretty late and she has to take a 1.5 hour train home (temporary) and wake up early. we don't have sex. I also wasn't in the mood atm but could have had it just for the sake of it, but she wasn't at all. I seemed a bit butthurt and gave her a passionate kiss when I left her at the train station out of ego/manipulative reasons.

she was also favorizing gym (not wanting to skip 2 days) over sex. she used to give blow jobs and always try to satisfy me, but it seems it will change. I think I shouldn't have gone to the gym with her, especially with no sex afterwards. I was probably manipulating her subconsiously wanting to give her some gym time with me to keep her as a sex source. I basically don't think i have what it takes to keep her around. yes, if I had abundance, I wouldn't care, but still I would have wanted to keep her around.

I've been a bit colder towards her after the situation and now she asks me if I'd like to come to her birthday. we know each other for 3 months and we hung out less than 10 times. I don't want her to think I'm her boyfriend but would like to fuck her more and eventually get to know her. any advices on how to proceed to keep her invested emotionally and letting her know that she needs to give me sex for it to work out without hurting her? she has a couple of red flags and I wouldn't want to marry her, but if I'm sexually satisfied, I'm also up for non-sex time. she seemed a bit more emotional the last time and I think she's about to get her period. it would be great if I could tell her to keep it fun without hurting her.

tl dr: I like her and I'd want to spend some non-sex time with her, but the main reason I'd accept other things is because of sex, and that alone is not enough to keep her, meaning if I straight up call her to hang out, she won't do it. we went to the gym together with no sex afterwards and now she wants me to come to her birthday after I've been colder than usual after our meet up. I need abundance and assertiveness, I'm working on it, but I'd like to know how to play this situation to keep it non-serious and letting her know I need sex from her without hurting her feelings.