Latin proper names certainly can be declined in the third declension, like the common noun rēx, rēgis
First, a small note on terminology: the noun rēx "king" is not generally considered to decline irregularly. Latin has several declension classes for nouns; I think you most likely are considering a noun to be regular if it belongs to the first or second declension, but the majority of nouns in the third, fourth, and fifth declension also follow rules according to their declension class and are considered regular in that case (although the third declension has several subclasses, and the fourth and especially fifth declensions are so small that the total number of nouns that follow their regular patterns is also fairly small.)
The third declension contains a great number of common nouns and adjectives in Latin. It also contains many well-attested proper nouns, such as Cicerō, Caesar, Iūnō, Mārs, which form their accusatives as Cicerōnem, Caesarem, Iūnōnem, Mārtem.
There are a number of examples of dog names from antiquity that are identical to Greek common nouns of the third declension, which you can see in the answers to What did the Greeks and Romans call their pets?
I haven't yet found an example of a dog named in antiquity after a native Latin common noun in the third declension, but I would expect that nothing would prevent this, and the name would decline the same way as the common noun. The answer by cnread on that page does mention the name Ferōx from a Latin third-declension adjective. It also mentions Tigris which is from the non-native common noun used in Latin for tiger, tigress.
I'm not familiar with examples of ancient Latin speakers changing the declension class of a Latin common noun when it is used as a name; if that ever occurs, I would interpret it as a form of suffixation with a suffix consisting of the first- or second-declension theme vowel. If the form Rēxum existed as a name, the expected nominative form would be Rēxus, with no form of the name being plain Rēx. However, Rēxus and Rēxum are very unlikely to exist as forms derived in Latin because derivatives of rēx are built on the stem rēg-, not on the nominative singular form rēx (which ends in a nominative singular suffix -s). An example of a suffixed form that is apparently built on the stem of rēx is the second-declension name Rēgulus, with the diminutive suffix -ulus.
There is no single rule on how to decline a name borrowed from another language
Joonas's answer mentions that if we consider Rex as a modern English name, there are a few possibilities for adapting it to Latin. This is not a matter of how Latin works, but of conventions for adapting borrowings between different languages.
With the names of individual humans, a convention may be preferred that preserves as much as possible the recognizability of the name while allowing it to be declined according to a standard Latin declension pattern. This might be why many non-Latin names are adapted to forms ending in first-declension terminations (-a in the nominative singular) for a female name, or second-declension terminations (-us in the nominative singular) for a male name.
However, this is not a requirement and other declension patterns are attested for borrowings.