Aye Ian,
Don't forget the engine is the same as used in the Virago XV250, insofar as I am aware. So carb parts listed for the XV250 may be suitable.
I have always used NRP for carb spares, here is the link:
NRP-carbs.co.ukI have also had a look at the Fowler's spares site and come up with this exploded diagram/parts/prices list and for item No.13 "Starter set #
3DM-1410A-10 Replaces 3DM-1410A-01" the price is £67.67inc. VAT and they have "No stock"!
Link to Fowler's page:
XV250-Virago-3dm5-1991-010-c/carburettorI had similar problems, with the choke plungers on both the SRX's and new ones cured the problem. The Honda Bros has always been cantankerous to start and even with new choke plungers, it can still be a bit "cold blooded", however, it is water-cooled and that might make a difference.
Hope this helps? Good health, Bill