Braden..... don't give up on the tube and syringe material delivery technique. You are headed in the right direction.
We used to use a similar technique to fill voids/delams in Nomex honeycomb panels used in turbine engine decks.
Just need a much larger syringe, a positive way to attach the tube to the syringe (small stainlesshose clamp would be ideal), and choose a different material for the hose, and possibly add a nozzle to the end of the hose.
A real silicone hose might be the ticket..... It would conform to and lay still in the fillet trough allowing you to simply pull withdraw as you dispense material.
Downside might be tendency to expand under pressure in tube if material is too thick.
Alternatively, for very high viscosity or general use, you could use a section of PTFE tube..... It is straight and stiff..... You could still put the tip right where you want it.....
Or you could hybridization it..... A PTFE wand of sufficient length connected to the syringe (fixed in a mount) with a flexible real silicone hose..... You can then manipulate the syringe with one finger while using the dominant hand to guide the wand along fillet area.
Just my 2/100ths of a $1 debt note.