Your current BB numbers in your 125g are in line with the bio load of the three fish you already have. As long as those three fish are the only ones going in your new tank then you should, in theory, have pretty much an instant cycle, or as close to it as you're likely to get.
If I was doing it i'd fill the new tank up, put dechlorinator in, and put a power head in to give it a good "swish" about for a bit. There's no real benefit of transferring old water from your 125, just use 100% new. Make sure it's up to temperature and then add your filter and fish pretty much at the same time.
Then just monitor your parameters, and fish. The only gliche that can happen sometimes is an ever so slight spike in ammonia but it won't last long. This can happen because, don't forget, your old tank has BB on every submerged surface, as well as in the filter. Your new tank WON'T have any submerged BB, not for a while anyway. So your BB numbers in your new tank, at first, will be slightly lower than your current mature 125.
You can always add a bit of bottled bacteria to help you along with this anyway. Good luck, and congrats on the upgrade