I just sized and measured a piece of unloaded Dominion .38S&W brass which measures 0.380" outside neck diameter.
I also checked an old WRA Co. loaded 38S&W round which measures 0.383" just below the roll-crimp.
I guess you slugged your bore and cylinder throats to determine 0.361" is correct for your revolver ??
I will no doubt get 'finger wagged, OMGd and tut-tutted' for this but what I would do (and have done in similar circumstances) is remove the de-cap rod and bushing from the Lee sizing die, and slowly and gently re-size the loaded round just far enough to allow it to chamber correctly and easily.
Then check to make sure the bullet is still held securely by the crimp, and if not, set your seating die to use the built-in roll-crimp just enough to securely hold the bullet.
I have never owned, used or felt the need for a Lee Factory Crimp die for 38S&W, but that's just my opinion.
I first started with a Webley 38S&W in 1969, have been reloading for about 45+ years, have owned several 38S&Ws including Webleys and Enfields, and currently shoot a 1934 Colt Police Positive 38(S&W) which has an 0.357" bore, which is very convenient as I also reload .38 Special and .357.