I don't think this is possible with the strong notion of boundedness I believe you want. If you're willing to assume something weaker, it might be possible, but we'd be butting our heads against things that are quite hard. Also, a caveat: in what I discuss below, I want to assume that if we call something an $L$-function, then it's sufficiently nice (e.g., automorphic). This is different than your terminology.

First, what's the "right" notion of boundedness? For an $L$-function $L(s,f)=\sum a_f(n) n^{-s}$ of degree $d$, we know that the summatory function $S_f(X)$ of the coefficients satisfies
$$ S_f(X):=\sum_{n \leq X} a_f(n) = M(X) + O(X^{\frac{d-1}{d+1}+\epsilon}),$$
where $M(X)$ is a term arising from the possible pole of $L(s,f)$ at $s=1$; if there is no pole, $M(X)=0$. We expect, though, to the best of my knowledge, we have no idea how to show, that the error can be improved to $O(X^{\frac{d-1}{2d}+\epsilon})$. Notice that, in either what we know or what we conjecture, if $d=1$, as is the case for the zeta function and Dirichlet $L$-functions, these bounds are $O(X^\epsilon)$, whence it's reasonable to ask whether $S_f(X)=M(X)+O(1)$. But if $d\geq 2$, then the error is bigger, and so it's probably not reasonable to ask for $S_f(X)=M(X)+O(1)$ (even if we tweak $L(s,f)$ suitably). However, we note that the conjectured error is always $O(X^{\frac{1}{2}-\delta})$ for some $\delta>0$; this is what I want to focus on. I suspect, though I'm by no means willing to conjecture it, that beating square root cancellation might imply automorphicity and hence RH (provided one adds some conditions so that it's not trivial -- e.g., require $L(s,|f|^2)$ to have a pole at $s=1$).

On to your specific question about "tweaking" an $L$-function. Given $L(s,f)$, if $M(X)=0$ (i.e., there is no pole at $s=1$), then, with the revised notion above, $L(s,f)$ itself works. If $L(s,f)$ does have a pole, it's easy to cook up a Dirichlet polynomial $g(s)$ that will kill the pole and not introduce zeros, and we can look at the series $g(s)L(s,f)$ and this will work with the revised notion. If we believe my suspicion, then this tweaking suffices to identify $L$-functions satisfying RH.

If you really want to get bounded summatory functions, I think you're out of luck (though I can't prove it, and from here on out, this answer will get progressively more and more wishy-washy). Let's say $G(s)$ is a Dirichlet series that has the same non-trivial zeros in the critical strip as $L(s,f)$. If $G(s)$ is reasonably nice (e.g., is in the extended Selberg class), then $G(s)/L(s,f)$ will behave a lot like a degree 0 element of the Selberg class. Such elements are known to be Dirichlet polynomials, and it would likely be possible, assuming $G(s)$ is sufficiently nice, to show that this is the case for $G(s)/L(s,f)$. A Dirichlet polynomial can kill a pole at $s=1$, but it's extremely unlikely that it'll cancel out the error terms (in essence, they are "legitimately" error, so they shouldn't satisfy any fixed relation as would arise from a Dirichlet polynomial). If $G(s)$ is not sufficiently nice, I don't know how to argue, except to say that there really shouldn't be any coherent way to force the error terms to cancel more.

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