First - talk to your doctor, as you note. Your pediatrician will be able to help answer your questions and guide you to find different options for helping address the root of your problem. Before you do, spend a week or two writing down everything related to your daughter's sleep patterns in a diary - when bedtime starts, how long it takes, how it went, how long she slept, how she was feeling when she awoke, when she naps, etc. That will give your pediatrician a lot of information to work with, and might help get a better solution.
Second - it sounds like you're in a "vicious cycle" here. Three is a common age for sleep problems; my first had significant problems around 2.5-3 and my second (33 months now) is starting to. My advice is to find a way to break that cycle. I don't know if the issues with waking up are related to sleep (and think your doctor is the one to ask that of), but if you're having so much trouble with sleep it's very possible that fixing the bedtime/sleep issues will help.
You say she's going to bed late and waking up late; find a way to shift that up a few hours if you can. This might not be possible, and it's not a huge deal until she's going to school, but if you can shift it up a few hours, I would. If she's getting enough sleep (ask your doctor about exactly how much, but typically 10 hours or so is common, though 12-14 is recommended at that age).
Otherwise, the main thing that helped us with our first was to pay attention to his daily cycles, and identify the sweet spot for bedtime.
Children (like adults) don't want to go to bed when they're not tired, but differently from adults (some, anyway...) they also have a hard time with bedtime when they're overtired. They lose the ability to control their behavior due to tiredness, and you end up with the goofy laughter, jumping on beds, running around the room, hitting mommy and daddy, etc. Both undertired and overtired can lead to very difficult bedtimes.
Instead, pay attention to identify when she's in between the two stages. For our oldest, we originally tried bedtime at 7:30; he usually was undertired then. We sometimes let it get a bit late - 9 or 9:30 - on days where we went to the library after dinner and such; he usually was overtired then. Overall, we found that for him, 8:30 was the sweet spot; about 80% of bedtimes went smoothly when we started the routine then. (This meant 10 hours of sleep, as he woke up 6:30-7:30 after falling asleep a bit after 9, plus naps.)
The other 20% we learned to identify days when they needed to be a bit earlier or later; this often coincided with days he either didn't nap, napped longer, or exerted himself more during the day. And sometimes we have "bad" bedtimes still - such is life, but at least it's now more like once every few weeks instead of every day.
If she's still consciously resisting bedtimes, changing a few things up may help. Change how you handle toys in bed (allow a toy or two if you didn't before, or make it more of a game, or perhaps get a special new stuffed animal and talk to her about it being to help with bedtime). Add some music into the routine if she likes music. Change the order of what you do during the routine - PJs then Toothbrush then Potty then Books, or Potty then Book then PJ then Toothbrush then another book. Whatever works to help her reset and get out of the vicious cycle you're in now.
