How to identify the GPD gene when the sequence varies between organisms?
I might be misunderstanding you here, but I take from your links above that you want to match the GPD promoter regions from two (distantly related) fungi Agaricus bisporus and Lentinula edodes, to the GPD promoter of Leishmania major, which belongs to a completely different Kingdom!
Promoter regions tend to be relatively poorly conserved between species compared with protein coding regions, even for closely related species. Given the evolutionary distance between the species you are mentioning, the chance you will find any homology between their promoter regions is probably zero.

Furthermore, how would I identify the GPD gene in another organism if I'm unable to compare it to a known sequence?

What I would do is to take the translated GPD protein sequence, which for Lentinula edodes would be GenBank BAA83550.1. I would then use that to search for protein matches using blastp, specifically subsetting for Leishmania major; and use the result to locate the coding gene in the genome. You can also do this in one single step with tblastn, which looks for matches in a translated nucleotide database (see this tblastn example query).
You can then simply take the 1000 bp or so upstream of the coding region to represent your GPD promoter.