I'm like you, one day with less nursing and my supply drops straight away. I have friends who say it takes weeks for their supply to drop even if one nursing is removed for good!
Given she had a tummy ache, I'd definitely want to avoid formula at this time. I also wouldn't assume you don't have milk for her, milk is produced when your breasts are being stimulated, I've found that my breasts can feel really empty, but my 5 month old can very quickly get a mouth full of milk and be dribbling it everywhere!
A 3 month old probably isn't quite such an efficient nurser, but she has to learn, you are also around the stage of transitioning from hormone driven supply, to supply and demand, so if she's been used to feeding of a full breast, she's going to have to learn that it isn't always going to be that way!
I understand the desire to sleep, I have narcolepsy and two other children, so even one bad night can really set me back, but if you can stomach it, I'd really try hard to just nurse her, both to avoid formula and for the long term good of your nursing relationship.
Obviously a lot of this depends on how you feel about formula, personally, it's the first bottle I really want to avoid, because I know that mentally once it's no longer exclusive breastfeeding, that I wouldn't feel such an incentive to maintain it. The first bottle my son got was necessary, I'd fallen down the stairs and they were worried about spinal injuries, so I was flat on my back waiting for an x-ray, no way I could move to feed him and he wouldn't have managed to get a good feed with me flat on my back (some babies would cope fine), but then a month or so later, when it was a choice of work really hard at nursing, or give him formula it was a lot easier than I liked to give in and give formula.
