As a side note, for your goals it seems like a “mule” configuration would be an option. This should reduce output losses but isn’t very suitable for general flashlight use. Similar to Dimbo The Blinky’s suggestion I think that this should be achievable with a “Convoy S2+”: just disassemble, remove the reflector, and re-assemble. That would be an 18650 (li-ion) configuration and it might require a small spacer for the battery due to the extra 10-15mm of battery tube gained when removing the reflector.
Moving on to AA stuff… for the output level you want to maintain (>300lumens) I think your budget and form factor are both making things tricky. I don’t see 1xAA maintaining 300 lumens - maybe 100 lumens would be more realistic. 4xAA in a cylinder form factor is very expensive (>$40). By busting your budget a bit you could pick up the Starry Light SA-22 for maybe $30 shipped(?), but it doesn’t meet your form factor requirements.
If AA is definitely the rule, what about an AA video light? Those should be available to use higher AA counts and hopefully produce a decent amount of light.
IMO 300 lumens into an umbrella isn’t a lot, but it depends on the situation of course.
EDIT: failing all of the above, another potential option would be something like two Hugsby XP-12 units (OrionLight - Hugsby XP-12 Review) attached to the umbrella. 2xAA is better able to maintain constant output across the discharge of your AA’s and two XP-12’s should get you in the ballpark of 200 lumens total. While I haven’t used Hugsby lights myself I understand that they have good output (as long as it’s a genuine Hugsby unit and not an imitation).