Instead of using UnitConvert, you can just divide the output by the relevant output unit. Here's one implementation of the idea, based on the OP's version:

ClearAll @ NoUnits;
SetAttributes[NoUnits, HoldAll];

NoUnits[fun_, quants : {__Quantity}, output : Quantity[1, out_]] := 
   fun/output /. Thread[quants[[All, 1]] -> quants] /. Quantity[x_, _] :> x

Usage:

LFullNU[l_, T_, a_, b_] = NoUnits2[Lfull[l, T, a, b], {Quantity[l, "Micrometers"], Quantity[T, "Kelvins"]}, Quantity[1, "Microflicks"]];

Seems to work. I have as inputs the full Quantitys to make the replacements easier.

As a check, you can try

BBNU[l_, T_] = NoUnits2[BB[l, T], {Quantity[l, "Micrometers"], Quantity[T, "Kelvins"]}, Quantity[1, "Microflicks"]];

which yields the same result as in the OP.

Instead of using UnitConvert, you can just divide the output by the relevant output unit. Here's one implementation of the idea, based on the OP's version:

ClearAll @ NoUnits;
SetAttributes[NoUnits, HoldAll];

NoUnits[fun_, quants : {__Quantity}, output : Quantity[1, out_]] := 
   Expand[fun/output /. Thread[quants[[All, 1]] -> quants]] /. x_ y_Piecewise :> PiecewiseExpand[x y] /. Quantity[x_, _] :> x

Usage:

LFullNU[l_, T_, a_, b_] = NoUnits2[Lfull[l, T, a, b], {Quantity[l, "Micrometers"], Quantity[T, "Kelvins"]}, Quantity[1, "Microflicks"]];

Seems to work. I have as inputs the full Quantitys to make the replacements easier.

As a check, you can try

BBNU[l_, T_] = NoUnits2[BB[l, T], {Quantity[l, "Micrometers"], Quantity[T, "Kelvins"]}, Quantity[1, "Microflicks"]];

which yields the same result as in the OP.