The OT doesn't question the validity of the marriage. IIRC the texts do not say their marriage was invalid.
They would certainly be welcome in the pews, as penitent believers, but not many pastors would be ready to perform the wedding ceremony.
Based on some of the comments from some of the Roman Catholics on these forums, I'd question if they would be welcome.
Would they be able to partake of communion in a Roman Catholic church?
Forget the Catholic Church. If someone murdered the husband of the woman he wanted to marry, in what church would the clergy perform the ceremony?
You could probably find one, no doubt, but most would decline.
God did not require David give her up, and in fact the bloodline of Christ goes back through Bathsheba. Honest repentance washes you clean. A clergyman who knew these two well enough to know the depths of their sorrow over what they did, might well agree to perform the ceremony. But most would not.
Granted that God seems to have looked the other way on the issue of multiple marriage in that era... but no orthodox Christian church will knowingly perform the ceremony for someone in a multiple relationship. So once you consider murder and multiple marriage, maybe David isn’t the best example for us.
Still, God uses imperfect people, and as was pointed out, despite his flaws his psalms have pride of place in Old Testament Scripture, and no one doubts David’s intimate walk with God. But doing his umpteenth marriage would still be a bridge too far for most pastors.