The definition of the Shimura lift of a classical cusp form is well documented. Zagier and Kohnen define a modified version of the lift for a cusp form $g(z)=\sum a(n)q^n \in S_{k+1/2}^{+}(4)$ in the Kohnen plus space for a fundamental discriminant $D$ with $(-1)^kD>0$ as

$$ S_D^+ (g(z)) = \sum_{n=1}^\infty \left( \sum_{d|n} \left(\frac{D}{d}\right)d^{k-1}a\left(\frac{n^2|D|}{d^2}\right)\right)q^n.$$

(Edit: Note that this definition is a slight modification of the original definition given by Shimura. It is defined only for forms in the plus space meeting the restriction $(-1)^kD>0$ and it includes a slightly different character than the original definition. The key consequence of this modified definition is that it lifts the form all the way down to level $N$, instead of just to level $2N$.)

They comment later in the same paper that the definition of the Shimura lift may be extended to modular forms in $M_{k+1/2}^+(4)$ by adding in a constant term of

$$\frac{1}{2}L_D(1-k)a(0)$$

to the above definition, where $L_D(1-k)$ is the L-series twisted by discriminant $D$ evaluated at $1-k$ by analytic continuation.

My question is the following: Is there a similar way to extend the definition of the Shimura lift to classical modular forms in $M_{k+1/2}^+(4N)$, in the plus space, for higher levels, with $N$ odd?

I have looked around somewhat in the literature; there are many sources which explain how to generalize the Shimura lift to cusp forms of higher level, but none seem to explain if/how you can extend the lift to non-cuspidal forms. If anyone knows a good source to read about this in the literature, I would appreciate being pointed in the right direction. If not, perhaps a summary of why the lift has not been extended this way/what the difficulties are.