With the veg fingers do they need to kind of dissolve when they gum them?
root veg will need to be cooked so that a fork goes right through easily rather than al dente I would say, just to avoid any harder bits, but it is not vital, with finger foods often they will just avoid or spit out the bits they don't like.
Mine really liked steamed asparagus, then green beans, they certainly do not dissolve when gummed but can be gnawed and sucked and generally stay enough together for LO to have a good go at them pulping them up before they try to swallow. Sugar snaps and mange tout were harder and he still struggles at times now with them, there is a long stringy bit which can cause DS to gag at 5yo unless we remove it or cut it up into smaller pieces (which is then not really the sort of finger food you want for a 6 month old as it is more of a bit than a wedge). Most kids seem to like chips - we oven bake various veg into "chips" which are great finger food (or potato wedges with skins left on)
You could try cooked apple or pear instead?
Apples are supposed to be brilliant for reflux although I thought they were quite acidic but it was more recently I discovered they are supposed to be good.
Mine couldn't have coped with dropping reflux meds at 6 months even though he was big on solids. I hope it doesn't cause any further problems.