Well, i do not know the answer in general but since you are asking for a reference and if

there are some conditions on $A$ and $B$ that guarantee these are the only non-trivial ideals of $A \otimes B$?

There are some related results, for example:
Let $B$ an arbitrary algebra over a field.

Every ideal of the algebra $A\otimes B$, where $A$ is a central simple algebra, is of the form $A\otimes I$, where $I$ is an ideal of the algebra $B$.

This is theorem 4.3.2, p. 74, from the book of Drozd-Kirichenko.

Maybe this reference might also be of some interest to you.

Well, i do not know the answer in general but since you are asking for a reference and if

there are some conditions on $A$ and $B$ that guarantee these are the only non-trivial ideals of $A \otimes B$?

There are some related results, for example:
Let $B$ an arbitrary algebra over a field.

Every ideal of the algebra $A\otimes B$, where $A$ is a central simple algebra, is of the form $A\otimes I$, where $I$ is an ideal of the algebra $B$.

This is theorem 4.3.2, p. 74, from the book of Drozd-Kirichenko.
(although i suspect this may not be of much use for the particular case you are interested in, as explained in the last paragraph of the OP).

Maybe this reference might also be of some interest to you.