Hi Michel,
You are correct. On p. 86 and p. 93 of
Jagdverband 44: Squadron of Experten, (2008), Robert Forsyth identifies Uffz. Franz Koster as a jet pilot from JG 7.
In Brown et al.,
Messerschmitt 262s of KG & KG(J) Units, (2010), p. 41, there is a photo of the original pasted strips of teletype of Hptm. Werner Gutowski's message to LFl. Kdo. 6 listing the last names of the pilots in JV 44 on 27 April 1944. At first there seems to be no mention of a Koster. However, there is a Uffz. Knier who I cannot identify and I suspect the telegrapher garbled the name Koster from a written list.
I also suspect that Joe Baugher took the name Peter Koster from one of these sites:
asisbiz.com/Luftwaffe/Luftwaffe-aerial-victories-1945.html
http://www.luftwaffe.cz/1945.html
I have no idea why Joe Baugher associated this loss with a Fw 190 D-9.
As to there only being one P-47 loss compared to five claims, unless there were badly damaged P-47s with unharmed pilots that limped back to base, this must be another case of over-claiming.
Jim