There is no quick fix that I've found -- however, after 2 years on a GAPS-style program, I've found that I can eat moderate quantities of many foods I couldn't eat before, with no real digestive issues.
In terms of meat, though, there are a LOT of options. If possible, I'd drop the dairy completely for a while, instead of substituting goat/sheep dairy. Meat options can include all kinds of things... fish, poultry (chicken, cornish hen, turkey, quail, pheasant, ostrich), goat, lamb, mutton (sheep), venison, rabbit, seafood.... there are so many options.
Next, I'd recommend a REALLY GOOD pro-biotic. There are a couple of excellent ones on the GAPS site, or you can check the refrigerator section of your local health-food store for a probiotic supplement with a minimum of 30 million bacteria, guaranteed.
It isn't fast, and it isn't easy -- but it worked for me.