I'm in the following situation. I have a self-injective finite-dimensional basic algebra $\Lambda$ (hence Frobenius) over a perfect field and two finite-dimensional invertible $\Lambda$-bimodules $M$ and $N$ which are isomorphic in the stable category of $\Lambda$-bimodules, i.e. there are finite-dimensional projective $\Lambda$-bimodules $P$ and $Q$ such that $M\oplus P\cong N\oplus Q$. I wish this implied that they were strictly isomorphic as $\Lambda$-bimodules $M\cong N$, but I'm not totally sure whether this is true, nor am I able to prove it. Is this known? Are there maybe counterexamples?
Here is an easy counterexample: let $\Lambda =F\oplus F$ (sum of 2 copies of the base field). Then any $\Lambda-$bimodule is projective, so everything is isomorphic in the stable category. However non-isomorphic invertible bimodules do exist: take $M=\Lambda$ and $N=M$ but with the right action twisted by an automorphism of $\Lambda$ permuting two copies of $F$. Thus some extra assumptions are needed..