There are reasons why this wouldn't have worked at all. I'll explain at the end, but first the numbers you ask for.
Liquid mass in Orbiter plumbing:
Common feedline (between external tank disconnect valve and engine
prevalve): 4000 lbm LO2, 250 lbm LH2
Engine feedline (between engine prevalve and SSME): 298 lbm/line LO2,
22 lbm/line LH2
Using the engine ...
The NASA document "Liquid Rocket Lines, Bellows, Flexible Hoses, and Filters" contains an extensive discussion of propellant line insulation in Chapter 220.127.116.11 Insulation.
Quick summary: Many types are used, but vacuum-jacketed hard lines with flexible joints were used extensively in the Saturn and Shuttle propulsion systems to maintain propellant quality. ...
My guess is that you don't have to insulate the lines, because the rate at which you charge and discharge your tanks is high enough that you won't gain enough heat to vaporize your Lox.
If your mass flow rate was really low,maybe you would have to consider the heat loss while in transit from tank to injector.