The side-valve JAP would share crankcase mounting detail with the correct OHV, making it easier to fit the correct motor when it comes along - and it will if you get the bike running and rideable. It's not correct, but not as incorrect as fitting a completely random motor. Engine plates, oiling, magneto, ... would be the same as the OHV.
The side-valve JAP motor was reasonably common and should be findable and inexpensive. The years 1920 to 1930 follow the word PNEUMATICS (P=1920, N=1921,...) so a good motor would be 1928-29 (I or C) with 1927 or 1930 possibly OK too. The 500 SV is usually K, so the engine number would be something like K/I 9999 for 1928 or K/C 9999 for 1929. The 350 would be J/I 9999 for 1928 etc.
Like S.A., Blackburne and others, JAP would supply motors for everything from lawn mowers to concrete mixers to railway inspection trolleys, so it's buyer beware as usual.
Leon