I'll come back to your specific question, Graham: whether the 3.5 hp Briggs 550 series engine would be a suitable direct replacement for the current engine. The answer is no, because the current engine has a reduction gear on its output shaft, to make the output shaft speed suitable for the mower. I'm guessing that the original engine was a thumping old English 4 stroke that did its best work at about 2,000 rpm. Modern equivalent engines, including the 550 Briggs, do their best work at 3,500 rpm, so you need an engine with a 2 to 1 stepdown gearbox built-in. Both Briggs and Honda make suitable engines, but you need to find one. The 3.5 hp output level is, in my opinion, barely sufficient for a 24" mower. You might find it a bit reluctant to climb significant slopes whilst mowing (that is a heavy mower).
The obvious alternative to an engine with a built-in reduction gearbox is to use a pulley on the mower drive shaft (not the engine output shaft) of double the current diameter, and achieve the required step-down entirely in the belt drive, but it is evident that there is not room for the larger pulley, so the reduction gearbox looks like the best option. Note that because your mower uses a belt drive between the engine output and the mower drive, the height of the output shaft from the engine is not an important issue (within reason): all that happens is that the length of the belts may change. However I do not see a mechanism for tensioning those belts, which may be a problem.
There are certain obvious attractions in reconditioning the old Honda engine, provided you do not need to buy many new parts (Honda spares are reputed to be expensive).
