Dog person here:
If the dogs are already adept at going over a 4’ fence, a 6’ fence may not hold them. People can inadvertently train their dogs to climb/jump higher and higher fences by only raising it 1-2 feet at a time. To truly be secure, their holding area should have a chain link or solid roof, without any gaps for the dogs to squeeze through between the walls and roof. AND a solid bottom or buried walls to keep them from digging out. You may be able to get away with skipping the roof if you raise the walls to 6’ and then add one of those toppers that leans in toward the dogs, so they can’t get good purchase as they clamber over. Coyote rollers are also effective for some dogs, but no guarantees, and again you risk just reaching them to be more skilled climbers/jumpers.
I’ll second the above poster to say that you shouldn’t fall into the trap of thinking of these dogs as “killer pit bulls.” There are good and bad dogs of every breed. Their current bad reputation is caused by media sensationalism, which causes people who want “intimidating” dogs to get poorly bred pits and not raise them properly, which adds to more media sensationalism, etc. In past decades other breeds suffered from the same stigma - Rottweilers and German Shepherds, for example. Not to downplay your specific situation, because obviously THESE pits are dangerous, but don’t assume that they all are.
I can’t comment much on building a secure goat shelter as my experience is with keeping dogs contained rather than out, except that the same principles would work - chain link or solid fence, secure roof, buried wire. I don’t know about the right spacing for electric wire, but I agree that you would want to make sure that it is a STRONG current. Dogs with serious aggression or predatory instinct will blast right through what is typically used as underground electric fencing in people’s yards.