I think that the matter of the paradox is that it causes explosion and trivializes the formal system, and if this explosion can be prevented, there is no problem.
I'm an unprofessional person but come up with an idea.
1.discard the general law of excluded middle and disjunctive syllogism.
Next, there are two kind of propositions. One is which must not contradict itself (namely, $P\wedge \lnot P$ can not be admitted), the other is which can contradict itself. An example of former is rigorous scientific claim and examples of latter are purely formal artificial statements, or sentences in ordinary contexts such as "I like him and I dislike him.".
2.admit law of excluded middle and disjunctive syllogism only for proposition of former kind.
3.the propositions "$X\in X, X\not\in X$" for the Russell set $X$ are of latter kind( namely, are artificial ones and don't appear in natural mathematical context), so doesn't cause explosion.
Now, I think that
4.in spite of both $X\in X\Rightarrow X\notin X$ and $X\notin X\Rightarrow X\in X$ are true, explosion doesn't occur because none of $X\in X$, $X\notin X$ and $X\in X \lor X\notin X$ is true (unless there are some extra axioms which prove them).
5.we can do usual mathematics because of 2.
Does this idea work?
How do you rigorously identify "artificial" statements?
->I define natural statements individually and identify artificial ones to the others. For example, one can assume statements which only relevant to sets in Gödel's constructible universe to be natural ones.
you may be trying to (re?)invent some form of paraconsistent logic.
->I don't know paraconsistent logic technically but I think that if we try to paste more than one scientific theories together or treat natural language, it is natural that there will be some contradictions and we have to treat them in some way. For this purpose, I discarded the full-strength explosion law.