Your approach makes sense, although you still may want to try removing the table names from the function:
=LookupFromPath(DomainName()&":"&ProjectName()&"/Data/"&CurrentDate("yyyy")&"/Budget Template Summary Transform",Version Quarter,Version Quarter,Value)
The name of the lookup table is part of the function's first argument, so it doesn't need to be repeated in the third and fourth arguments. (Similar syntax-wise to the more common Lookup() function).
Also, a single LookupFromPath() function can only look at a single time period, not a range of time periods (and, unfortunately, not an entire fiscal year all at once).
But it would be possible to creatively construct a long formula which sums results from two or more LookupFromPath() functions, each one looking at a specific month.
For example:
=LookupFromPath(DomainName()&":"&ProjectName()&"/Data/"&"January:2015"&"/Budget Template Summary Transform",Version Quarter,Version Quarter,Value)+LookupFromPath(DomainName()&":"&ProjectName()&"/Data/"&"February:2015"&"/Budget Template Summary Transform",Version Quarter,Version Quarter,Value)+LookupFromPath(DomainName()&":"&ProjectName()&"/Data/"&"March:2015"&"/Budget Template Summary Transform",Version Quarter,Version Quarter,Value)
Or a version which auto-adjusts to the currently selected year but stays in Q1 (assuming Q1 is Jan/Feb/Mar):
=LookupFromPath(DomainName()&":"&ProjectName()&"/Data/"&"January:"&CurrentDate("yyyy")&"/Budget Template Summary Transform",Version Quarter,Version Quarter,Value)+LookupFromPath(DomainName()&":"&ProjectName()&"/Data/"&"February:"&CurrentDate("yyyy")&"/Budget Template Summary Transform",Version Quarter,Version Quarter,Value)+LookupFromPath(DomainName()&":"&ProjectName()&"/Data/"&"March:"&CurrentDate("yyyy")&"/Budget Template Summary Transform",Version Quarter,Version Quarter,Value)