If a husband or wife is the only one to witness their spouse's death, can they lawfully remarry?
If the spouse is actually dead, then nothing legally prevents the widower from remarrying:
39 A wife is bound by law for as long as her husband lives; but if the husband is dead, she is free to be married to whomever she desires, only in the Lord. 1 Corinthians 7:39WEB
The only person whom the widowed spouse needs to convince (in a de facto sense) is their prospective spouse. And presumably, if that spouse didn't trust him/her to tell the truth on this issue, then they would not be marrying this person.
In Biblical law, there is no requirement for civil government "permission" to marry. Biblical civil government is reactive to crimes against the marriage covenant, not pro-active.