We started to give formula, at about 6 months, as dd still wasn't sleeping through, waking several times a night - I had gone back to work, and was dead on my feet. It made no difference whatsoever. DD Started sleeping through finally about a month ago, she is 10 and a half months.
The only thing that prompted this was me reducing the feeds I gave her, from about 3am onwards I wouldn't feed her until morning, and eventually she learnt to drop off back to sleep without a BF. I do think that for some babies, maybe hunger is the issue, but really formula isn't going to make a huge difference, in my opinion, the biggest thing that affects sleep is the ability to self settle - DD had to learn how to do this, she would wake and the only way she could get back to sleep was to feed. I started to feed her before she had a bath, or was dressed for bed, so she didn't fall asleep whilst feeding, and at night I gradually reduced the time I was feeding her, so she had chance to catch up in the day - have to say though, didn't do any of this until she was on solids, at 6 months old. she didn't want them before that, we offered her food, but she didn't know what to do with it! Loves her food now though, never stops eating, and still feeds 3 times a day...however, even now, I wouldn't say that how much she has to eat has any effect on her sleep, its more likely to be how late she went to bed that will make her wake, or teething.
It is so hard, sleep deprivation is the worst, theres a reason that they use it as torture! TBH, if the formula doesn't help the sleeping, you are going to be awake for longer anyway, as you will be making/warming bottles!
Can you get someone to give the feed early on, go to bed early for a few nights, then take the early morning shift? Even if you used EBM, and then at least you aren't missing a feed - that is, if you are determined to carry on feeding yourself - I don't know how much affect dropping one or two feeds would have on supply, but maybe it would be established by now, so you could use formula - I have nothing against using it - I do still now, for DD during the day, as I find she is less distracted with a bottle.
It;s so hard to decide what to do, if you don't get enough sleep its even harder - just remember it wont last forever though.