The answer is in general no - K(Rtext−mathrmMod) can fail to be well generated even when R is artinian. As you mention K(Rtext−mathrmMod) is compactly generated if R is of finite representation type. It turns out that the converse holds. This is a result of Jan Šťovíček which occurs as Proposition 2.6 in this paper. The precise result is:
Proposition Let R be a ring. The following are equivalent:
(i) K(Rtext−mathrmMod) is well generated;
(ii) K(Rtext−mathrmMod) is compactly generated;
(iii) R is left pure semisimple.
In particular, when R is artinian this occurs precisely when R has finite representation type.
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