I think you're going to have to discover how that thing communicates
"without modulation".  ;-)  I suspect you're referring to the 38KHz signal
as the "modulation".  (Probably should call it the "carrier").  But any
attempt at straight-through analog amplification is going to be plagued by
the noise from stray lights as you pointed out.  So you're going to have to
discover what the receiver is looking for, and detect, and repeat that, much
as the previous scheme did.

So, maybe it uses baseband codes without a carrier?  My first IR control
from a TV made in 1976 by GE used discrete frequencies in the 40-50KHz
range.  So the receiver had to decide which frequency you were throwing at
it.
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