Trains to Stirling on the Sunday morning are in short supply. The earliest arrival from the south-east is the 0934 Edinburgh to Dunblane, which gets into Stirling at 1030. Possibly too late, but at least it's a direct train. There just aren't enough early trains to find a train/bike/train option, such as riding from Falkirk High to Falkirk Grahamston (and then you'd be as well catching the FKG train from Edinburgh in the first place...), and connections via Croy/Lenzie aren't options at that time of the morning.
The 0830 from Edinburgh to Glasgow Queen St is normally sufficiently quiet to not have to worry about getting your bike onboard, but aim to get on at Waverley rather than Haymarket. You have 10 minutes to get from platform 3 to platform 2 once at GLQ. The 0937 to Stirling will usually be quiet, but may be busier if it's a big bike event.
If it were a social ride I'd be asking the organisers to wait a few more minutes for the east coasters. A good organiser will keep a watching brief on public transport and adjust accordingly.