Refuse to seat the delegation from Illinois if they go through with this.
Each house can choose to accept the results from each state or reject them. A Democratic-controlled House is not going to throw out the votes that support a Democrat candidate. A Republican-controlled House can’t throw out disputed votes that support the Democrat because of charges of racism, sexism, homophobia, islamophobia, transphobia, etc etc.
As you point out, Republicans cannot get on the ballot for important elections in California, like the last two Senate elections. They don’t even allow write-in ballots. One would think that they wouldn’t be so blatant when they achieve power, but look at what they’re doing in regards to abortion and gun rights. They just don’t care what a large portion of America thinks. They may come to care, but right now they don’t.
You are certainly correct about how things are supposed to work. They aren't working anymore.
Disallow the electoral votes from Illinois, and refuse to seat their elected representatives in the House and Senate. Apply the same "rules" to California and New Jersey. That is hardball.
If the Democrats are going to change the "laws" to suit themselves, we might as well do the same. There is no point in playing by their rules anymore.
They will never care and cannot be intimidated. Only their replacement can (maybe) change anything.
It would be nice if that replacement could be done by honest elections.