Tested (albeit, non-rigorously) BrowserUK's with a data file (very modestly varied from OP) eq:

>DL;H1_ENSP00000194530_chr2_202024 CCCC---GCCTTCTCGCTGCCCAGC--CCCGGGGA
+GGGAGG*
">DL;H2_ENSP00000194530_chr2_202024 CCCC---GCCTTCTCGCTGCCCAGC--CCCGGGG
+AGGGAGG*" line 2
>DL;H1_ENSP00000194530_chr2_202024 CCCC---GCCTTCDLTCGCTGCCCAGC--CCCGGG
+GAGGGAGG* line3
>DL;H1_ENSP00000194530_chr2_202024 CCCC---GCCTTCTCGCTGCCCAGC--CCCGGGGA
+GGGAGG line 4
>DL;H1_ENSP00000194530_chr2_202024 CCCC---GCCTTCTCGCTGCCCAGC--CCCGG*GG
+AGGGAGG line 5
[download]

and output is:

>H1_ENSP00000194530_chr2_202024 CCCCGCCTTCTCGCTGCCCAGCCCCGGGGAGGGAGG
">H2_ENSP00000194530_chr2_202024 CCCCGCCTTCTCGCTGCCCAGCCCCGGGGAGGGAGG"
+ line 2
>H1_ENSP00000194530_chr2_202024 CCCCGCCTTCDLTCGCTGCCCAGCCCCGGGGAGGGAGG
+ line3
>H1_ENSP00000194530_chr2_202024 CCCCGCCTTCTCGCTGCCCAGCCCCGGGGAGGGAGG l
+ine 4
>H1_ENSP00000194530_chr2_202024 CCCCGCCTTCTCGCTGCCCAGCCCCGGGGAGGGAGG l
+ine 5
[download]

Nice, BrowserUK; ++

Update: Fixed the mis-attribution. Give BrowserUK another ++ and I'll do penance in the dungeon; the more so, since it was he who answered a brain_dead question about his code.