This is my first time answering on this site, but your post really grabbed me. I am answering with the background of being a mum of a 6 month old boy, and a paediatric doctor in Australia.
Firstly, it sounds like things are really tough at the moment. Hopefully you will get some good sleep soon to recharge. I have a few suggestions, and will separate them out.
1) Share the load. I'm not sure if your baby is sleeping in your room or in another room, but arrange with your partner a system whereby one person is "on duty" for a certain part of the night, and the other gets some good quality sleep for a couple of hours. If your baby is in your room consider sleeping in another room (even the lounge room) so you get some uninterrupted sleep when you aren't "on duty". Having your partner take the first shift, and giving a bottle if needed will give you a good stretch straight up. Also, since she seems to be a good daytime napper, if you are still at home try sleeping when she does during the day. It isn't ideal for the long term but while things are crazy it may help you get through. Also, utilise your support network (family, friends, neighbours - anyone you trust) to come over and give you a hand. Maybe they can do the house work while you get a bit of happy playtime with your baby during the day (to remember that she isn't always crying) or visa-versa.
2) Total sleep volume. From your post it sounds like your baby is getting between 2.5 and 4.5 hours of day time sleep each day, which is great day time nap volume. Most three month old babies average between 10 and 16 hours of sleep per 24 hours, with the vast majority getting 12-14. If she has a natural sleep volume of 12 hours per day (for example), she will therefore be asleep for only 7.5 to 9.5 hours overnight after day time naps. So perhaps starting the night sleep later will give more time for YOU to effectively sleep while she is down. The number of 3 month old sleeping from 7.30pm through the night is small, and the parents are very lucky who get them! Perhaps try pushing back "bedtime" to closer to 9 or 9.30pm, and trying to get into bed yourself when she goes down. Then if she does have a 3 hour stretch straight up you will have gotten that too.
3) Eating. You said she is a mix of breast fed and bottle fed, but haven't specified how many of each feed type she takes over the day and night. It may be that she has become used to taking lots, or even the majority, of her daily milk intake overnight. At 3 months she is able to get used to drinking more during the day than night, but this may take some "training". To encourage more milk intake during the day, try to decrease night intake gradually. If she is "snack feeding" overnight (smaller, more frequent feeds) start by consolidating the feeds into 2-3 feeds only, by stretching out the gaps between feeds to a minimum time. For example, if she is snacking every hour or less start with perhaps minimum 1.5 hours between feeds for a few nights, and gradually stretch out to 3-4 hours (so eventually you are only getting up 2-3 times per night to feed). This does need to be very gradual, and try only stretching feeds out a further 15 minutes every 3-4 days. As she feeds less overnight she will start eating more during the day, but it will take time for her to readjust.
4) Making sure she recognises night vs day. Keep the room lights low, try to speak in a quiet voice and don't play games or be too fun when she wakes overnight. Easier said than done when you have a screaming baby, I know.
5) Settling. Besides feeding (which is very easy to fall back on every time she wakes overnight, but snack feeding can create a problem as outlined above) you need to work out what helps your little girl. Have a barrage of settling "tricks" already thought out to help settle her and to help keep you sane. Even if it may not settle her instantly, having something you are trying can stop the feeling of helplessness and frustration a little. I personally am a huge fan of the "Happiest Baby" settling technique (5 S's which you can google) - but this can stop working around the 3-4 month age. However, it may still work a treat, and reviewing the techniques may help. Consider a dummy (pacifier) if not already using one. Consider writing a list of things to try and sticking it in the room for when you are fuzzy headed overnight.
6) Sleep associations. Create positive sleep associations which will cue her at the start of each nap/sleep time and when she wakes overnight that it is sleep time. For example: swaddling, pacifier, white noise, small breathable snuggy (aka lovey or another small toy that cannot suffocate the baby). If she is sleeping better during the day than overnight try to work out what is different between the two (ie. light levels, noise levels etc).
7) Comfort. Ensure she isn't too warm or too cold overnight. Be aware that overheating is a SIDS risk, as is loose bedding. She shouldn't feel hot and sweaty to touch. Altneratively, if her face is cool she may not actually be too cold - slip a finger down her suit to feel her chest, if it is warm then she is just right. Don't judge if she is cold on her hands or feet as these are often cooler. Consider getting a room thermometer can experimenting with different amounts of clothing/bedding for different temperature ranges. There are baby monitors with temperature alarms if you think that may be an issue, but it is an expense you may be able to do without. Also, don't forget to make sure it isn't a nappy (diaper) needing changing or if she has vomited on herself etc.
8) Work on self settling. It sounds like she might need you to help her go back to sleep each time she wakes overnight (infant sleep cycles are about 45 minutes long). Try to put her in her bassinet/cot when she is settled and drowsy but not actually asleep. If she falls asleep in arms, jostle her on putting her to bed so she wakes up and sees she is in her bed (but hopefully not wake up so much she will get distressed). She will eventually get used to falling asleep in there and not need to cry for you to help put her back to sleep each time. Alternatively, cuddle/rock/feed her to sleep each time, but you will then have the issue of her needing you to go back to sleep at each sleep cycle. If you want to co-sleep make sure you read and implement safe co-sleeping practices. I personally do not recommend or think co sleeping is safe below a minimum age of 6 months due to SIDS risk, but this something you need to decide as parents.
Quite a long answer, but I hope this helps. Good luck. I'd love to hear back how you go.