tick bite?? help!

Hey y'all
My younger sister (does not have lyme) was outside and was bitten by an unknown bug on the back of her neck 2 days ago. She only saw the bug for a sec when she brushed her hand away and it was small and dark. She'd washed our dog 2 days ago and found a tick on our dog's ear (not uncommon) that we removed. She had been in the woods with our dog earlier that day too. Of course it could have been a different type of bug that bit like a fly, but i'm skeptical.
Now, the area where she was bitten has a red, bumpy appearance rash, but just like a "blob" not a target shape of erythra-migrans rash. There are two VERY swollen, like hard, painful to touch, visibly swollen, lymph nodes right by the rash. Her arm on the side she was bit is also a little numb/tingly and she had a bad headache that evening.
I had her go right to the dr's but he refused to give her antibiotics since she a) doesn't know it was a tick that bit her b) not a typical tick-bite rash c) no systemic symptoms like fever, joint pain, etc.
Does this sound like a tick-bite to y'all? I don't have a picture to post unfortunately.
What would you do in this circumstance? Are there different types of bugs that could cause this bite? Do you think it's worth pursuing a different doc to get antibiotics (i've heard doxy 200mg twice a day for 30days is safest). Or is there some "time" since this is a fresh bite, to wait for other symptoms to show?
I've been sick with lyme for 3 years now. I wouldn't wish this on my worst enemy let alone my sister!

Thanks everyone for responses! Sister went to our regular MD who unfortunately is same one to refuse to even test me for lyme. He refused to treat. She started to feel worse & took her to urgent care, they pulled something out of the original bite, a little black "stinger" or I think potentially part of the tick which was left when she accidentally pulled it off herself. They also gave her doxy 100mg 2x/day for ten days. Not the best prescription but our area is notoriously bad for being lyme-aware, and it's better than nothing. Hoping we can get prescription refilled after 10 days to continue tx. Also hoping to get her into my LLMD but unfortunately he is in Pennsylvania and traveling is so hard rn due to COVID-19.
One more question - has anyone had EXTREME swollen lymph nodes after bite? hers were swollen by the first day after, now they are HUGE and painful, visibly swollen, I'd say the around the size of a large marble or even bigger, maybe strawberry size! On the bright side, her rash went away immediately after they removed whatever was still in the bite.
Still concerned for her but also glad that i suffer with lyme so i know how important it is to treat and could recognize the s/s in her right away. Our sufferings do have some merit- at least we can help others!