On the other hand I know of at least one gentleman who has converted to Sri Vaishnavism and has kept his Christian name. Do scriptures say anything about this?
Yes, they do. And here, that person has retained his birth name which is alright.
During Agamic initiation, the Guru gives the disciple a name. This is known as the DikshA NAma (the initiation name). This indicates that now you are in our Sampradaya and also indicates the fact that the disciple has obtained a new birth (hence a new name).
This happens at the end of DikshA.
Evam idam dikshAtrayam nirvartya pashchAt tasmai ishtam mantram dadyAt | Tato guruh shishyashirasi swacharanau niveshya ishtamantra kramopayuktAn sarvAn angamantrAn tasminneva kAle kramena vA yathAhadhikAramupadishya swAngeshu kimpyAngam shishyam sparshayitvA tadangamAtrikAksharAdim dwyaksharam tryaksaram chaturaksharam vA AnandanAtha shavadAntam tasya nAm kritvA dashama khandoktAn AchArAnanushishyAt ||
From Nityotsava's 2nd chapter, called DikshA Krama's sub-section called IshtamantradAnam.
In this manner the Guru should complete the three initiations and give the disciple his Ishta Mantra. Thereafter, the Guru, placing his foot on the disciple's head, shall instruct him with all the related Anga Mantras either during that time itself or later, in succession. Now, the Guru should make the disciple touch one of his limbs. Noting the MatrikA Akshara (one of the 50 Sanskrit alphabets), NyAsa of which is done during MatrikAnyAsa, on that particular limb (part of the body) touched, the Guru shall frame a name starting with that alphabet, comprised of two, three or four alphabets. And, upon adding the the suffix "Ananda NAtha" to the name thus obtained, he thus gives the disciple a new name. Thereafter, he instructs the disciple with the conducts that are described in ParashurAma Kalpasutara's 10th chapter
To know how this MAtrikA NyAsa is done, you can check this page
So, we know that on the left knee, for example, the Sanskrit alphabet Da (द), is to be Nyasa-ed with. So, assume the disciple touches the Guru's left knee.
Then the Guru has to frame a name starting with Da, comprised of two, three or four alphabets. Let's assume the chosen name is "DayA" which is meeting all the criteria. So, the DikshA NAma or Guru given name for that disciple will be DayAnanda NAtha.
(Note - Here, the suffix Ananda NAtha is peculiar to the Sri VidyA sect and it will change with sect).
But, this is briefly the process of renaming the disciple.
After the initiation, the disciple, at least in the religious circle, is known by his DikshA nAma more but he can retain his birth name and use it.
For example, BhAskara RAya has retained his birth name while writing many treatises where as his DikshA NAma was BhAsurAnanda NAtha.
So, it can not be said that the Christian person is at fault however he must have his own Guru-given name too.