Two things have to be noted:
I am not an Elvish language expert, but I am unaware that the meaning of Gothmog can be rendered as 'voice of Morgoth'. From the Etymologies in HoME V, reflecting the status of the Elven languages at the time of the writing of LotR, the name can be translated from the two stems
GOS-, GOTH "dread. Q osse terror, as name Osse. Cf. Mandos (see MBAD). N has Oeros for Osse (*Goss). Cf. Taur-os [TAWAR]. N gost dread, terror; gosta- fear exceedingly; cf. Gothrog = Dread Demon [RUK]; Gothmog [MBAW]. Gostir 'dread glance', dragon-name [THE]." (HoME V Etym.)
and MBAW- "compel, force, subject, oppress. Q mauya- compel; mausta compulsion; maure need. N baug tyrannous, cruel, oppressive; bauglo to oppress; bauglir tyrant, oppressor; bui (*mauy-) (impersonal), baur need. Cf. Gothmog (*Gothombauk-) [GOS]." (ibid.)
Even if you equate S./N. bauglir with Morgoth, the name would only mean something like 'dread of Morgoth'.
Much more importantly, LotR states the following:
"The Lieutenant of the Tower of Barad-dūr he was, and his name is remembered in no tale; for he himself had forgotten it" (LotR V, 10).
This logically also excludes the possibility that Gothmog was not the Mouth's real, original name, but only a (later) title. Had it been that, it would have been remembered in the very same tale in which he appears.
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