You'll be much happier with a GX390 or GX340 Honda over the Briggs. The shaft size is 1"x~3.5" (normally listed as 3 31/64) Used Honda GX340 or GX390 are also easy to find with this shaft size. It is the most common ("Q" types)
Several years back we tried the same truckloader, but the preceding TR1103 model vs your TR1104. Nothing but problems with the 11hp B&S I/C, not unlike nearly every other application we have had this engine on (the 8 and 10hp versions as well - 'newer' with the plastic tanks) Fuel pumps every year, constant carb problems, bent rods, and on high continuous load applications (e.g. push blowers) they would just eventually throw the rod through the side of the block. Some people have zero issues, one too many nightmares for us - Hondas only from now on. Have yet to 'kill' a GX honda even with some of our pushblowers exceeding 1500hrs of full throttle constant load use.
A GX390 (or a 340) will fit on the Billy Goat frame as-is, it was available from BG as the TR1304H. I actually have a picture of our old TR1103 from when it was sold, see attachment. You can see the space between the muffler and the blower housing is pretty tight, and the Honda GX muffler protrudes much further. Billy Goat used a spacer block between the engine mounting face and the blower housing to get more clearance between engine/muffler and housing on the Honda and Vanguard briggs models. IIRC the frame will be pre-drilled with engine mounting holes further back already for these applications.
You will need to do the same with the spacer(s) and the rearward mounting holes. As far as I can tell the standard 'big' Honda muffler with heatshield was still used, so that should be it. But, if you still have clearance problems with muffler even after spacing engine back you can always swap to the smaller Honda GX270/290 oblong style muffler/heatshield and have a ton of clearance. Again, should not be required.
Despite upping the power, there will be little performance increase with the Honda on your Billy Goat, the 8" intake is the weak link of this machine and clogs up like crazy. Reliability/Dependability will definitely be much better with the Honda, though. Gotta love easy starting on the first pull!
If you already are dealing with a lot of clogging, for the cost of upgrading the engine you may just want to keep your eye out for a used truckloader with a larger engine and a 10"+ hose and sell your current unit to offset. If you find a 16hp or 18hp HTR or TR Billy Goat you can even re-use your existing mounting setup and any discharge extensions.
If you don't have clogging problems and/or want to keep your unit, then swap on the Honda and call it a day.
Oh - one more thing. If you do go ahead and buy a Honda engine make sure you do not get a "Cyclone" air cleaner model. It won't fit within the Billy Goat TR/HTR frame without swapping to the standard GX340/390 air cleaner base and cover.
Billy Goat TR1103 engine to housing clearance;