I call bullshit on there being painkiller mixed in with antibiotics–never seen that done, usually antibiotics and painkillers are administered in two separate shots.
Bute is generally the go-to painkiller for horses. Stronger stuff is really rare unless an injury is severe or a there are extenuating circumstances.
When I was working at a breeding farm, one of the broodmares who had had a freak pasture accident and was on stall rest forever was on bute and omeprazole (to protect her stomach) for ages. She finally healed up and about a year later got a really bad abscess. The vet was able to try something new with her and put a fentanyl patch on her. Shaved down a spot of hair on her neck and stuck it on like a nicotine patch. Worked WONDERS for this mare, who was spending most of her time lying down prior to that (as of course the abscess was in her bad foot/leg she had injured previously).
This was close to 10 years ago, but it was fairly new at the time. The owner of the mare wanted to know why the vet hadn’t done it for the mare’s previous injury, and the vet said they didn’t have it at that time.
All that being said, a lot of vets don’t provide adequate pain medication for horses in my opinion. They seem to think that horses should just be able to suffer through it, or maybe think the pain isn’t that bad, because horses are such stoic creatures. I have watched equine dentists dig out teeth without pain relief/numbing (just because the horse is sedated does NOT mean they can’t feel it) and I was disgusted. I promptly hired a vet who specializes in equine dentistry and who uses novocaine/lidocaine before extractions. My vet provides pain relief as par for the course when a horse is injured, though thankfully (knocking on all the wood around me) I haven’t needed anything other than bute, banamine or previcox yet.
You could push your vet for pain relief but be prepared that they may not give it. I know plenty of old-school vets in my area that would brush an owner off completely in a case like this or tell them to give the horse bute for 3 days. At least with an abscess, if it is draining, the horse should get a lot of relief pretty quickly.