Their construction (Burgos-Feliu) carries over - direct - to any regular schemes, see e.g. the first lines of chapter 4, p.23 of their paper.
There are nevertheless several blockers to go further :
There is a lack in the litterature for Bloch’s higher Chow groups over arithmetic rings. I don’t think this is a serious problem.
To define intersection product for regular schemes one need to extend the Gillet-Soule moving lemma for K1 chains. Again I think it’s doable. For smooth schemes there is of course Levine’s approach (going back to Fulton).
In the end, proving that the constructed theory fits into the general picture of higher Arakelov groups. This is probably difficult. I don’t know what to do for general regular schemes. This is the main blocker in my opinion.
Note that using cohomological theories (as yours) one gets the second point essentially for free, as for homological theories (like Chow groups) you need lots of work and some additional hypothesis to define intersection product. That difference must be sensitive when comparing both types of groups.
nLab page on Maillot on higher arithmetic Chow