Let $\mathbb{K}$ be a fixed field. Take a finite-dimensional $\mathbb{K}$-algebra $A$ and look at the category $\text{mod}(A^e)$ of finite-dimensional $A$-bimodules. It has a full subcategory $\text{lrp}(A^e)$ that has a objects those $A$-bimodules which are projective when considered as left and as right $A$-modules.
Is that subcategory a dense (or even absolutely dense, or colimit-dense) subcategory of $\text{mod}(A^e)$? Is it in some other sense a 'generating subcategory' of $\text{mod}(A^e)$?
In Left-right projective bimodules and stable equivalences of Morita type it is claimed that if $A$ is a self-injective algebra, then, 'except for trivial cases', the complement $\text{mod}(A^e)\setminus\text{lrp}(A^e)$ has infinitely many pairwise non-isomorphic indecomposable objects. Don't know if that helps.
Yes, the subcategory $\operatorname{lpr}(A^{e})$ is dense in $\operatorname{mod}(A^{e})$.
The following lemma is an immediate consequence of Theorem 5.13 in
Kelly, G. M., The basic concepts of enriched category theory, Reprints in Theory and Applications of Categories, No. 10 (2005).
Lemma. If $\mathcal{E}\subset\mathcal{D}\subset\mathcal{C}$ are full inclusions of categories, and $\mathcal{E}$ is dense in $\mathcal{C}$, then $\mathcal{D}$ is dense in $\mathcal{C}$, and $\mathcal{E}$ is dense in $\mathcal{D}$. $\square$
Since $A^{e}\oplus A^{e}$ is an object of $\operatorname{lrp}(A^{e})$, to prove that $\operatorname{lrp}(A^{e})$ is dense in $\operatorname{mod}(A^{e})$ it is sufficient, by the Lemma, to prove that the full subcategory containing the single object $A^{e}\oplus A^{e}$ is dense in $\operatorname{mod}(A^{e})$.
But more generally, for any ring $R$, the single object $R\oplus R$ is dense in the module category of $R$. See, for example, (2.2) in
Isbell, J. R., Adequate subcategories, Ill. J. Math. 4, 541-552 (1960).
(Note that Isbell uses the term "left adequate" instead of "dense".)