Q-curves and twisting

It looks like your notion of strong $\mathbb{Q}$-curve over $K$ is what Peter Bruin and Andrea Ferraguti refer to as a $\mathbb{Q}$-curve being completely defined over $K$. Such curves have $L$-function factoring as a product of $L$-series of newforms for $\Gamma_1(N)$. This then seems to coincide with the definition of strongly modular given by Xevi Guitart and Jordi Quer. This latter set of authors provide an explicit example of an elliptic $\mathbb{Q}$-curve (which they call a building block after Elisabeth Pyle's thesis) over $K = \mathbb{Q}(\sqrt{-3})$ which is not strongly modular, and state that no curve isogenous to it over $\overline{\mathbb{Q}}$ and defined over $K$ can be strongly modular:

$$ Y^2 = X^3 + 4aX^2 + 2(a^2 + b\sqrt{-3})X, $$

for $a,b \in \mathbb{Q}$. I haven't checked the details, but this might give you what you're after?