I think it's pretty normal to have a delayed reaction like that. Did you cook the hamburger yourself? Have you eaten out somewhere that could have brushed the burger with butter or cooked it in "whirl" a buttery grease?
Formula wasn't an option ($-wise) for us, and I am thankful because it made me have to continue. I did consider formula for dd when she was 10 months old, I was 8 weeks pregnant and not producing much milk. The Alimentum is nasty, imho, she wouldn't drink it and I wouldn't either. Most milk protein intolerant babies can't handle it - it still has milk proteins they are just "pre-digested" If soy is ok, soy formula is an option, otherwise you'll have to go with the hypoallergenic Neocate or similar, which means big $$$ bucks.
My sister is milk-free too, nursing her 9 month old, and she tried reintroducing milk last month. Cora broke out in a rash, had visible blood in her poop, mucus, woke up hourly, and got a double ear infection. When I look back on how fast it went - I just have to keep saying "it was so worth it" to give them health, and a chance at life without food allergies.
Even if you have a few bad spells now and again, are most days good?