The run circuit will get its current from two places. With the side stand up, it comes through the side stand switch. Sounds like yours is working ok.
With the stand down, running current must come from the neutral switch which energizes the middle relay (called the "sidestand relay", it is not energized by the sidestand switch, but the neutral switch on the left side of the gearbox under the starter motor).
Your trouble is likely the neutral switch, its connection, or the relay and /or its connections.
First, locate the rubber boot under the starter with only one wire. Pull it back, slide off the blade connector, clean it, tighten it, Deoxit, and maybe apply a thin coating of waterproofing grease. It should slip back on the switch with some resistance. Retest your starting with the stand down in neutral.
No joy? Carefully pull the middle relay straight up without twisting or wriggling (this could spread the connectors). Replace with a known good relay. OMRONs make me happy, but you could simply exchange it for another in the fuse block and retest.
EDIT: Also check the adjustment for the sidestand switch. It is a simple carriage head bolt with a lock nut. Make sure it completely depresses the switch button in the down position, but no more and no less.