Instead of forcing the government to shut down, the GOP should make the compromises on the budget that they need to - as long as they don’t hard-write in any new or even old entitlements. Let those agency-defined payoffs for votes remain.
Then, using Impoundment, stop the slow bleeding of the Treasury by putting a stop to unpayable debt. Trump can just refuse to spend budgeted money in selected areas.
Now, this is best done with a simultaneous change in The Center for Medicare Services definition of “Eligible for government services”. Requiring that the agency change that to NO ILLEGAL ALIEN, EVEN IF PAROLED IN THE USA, WAITING DEPORTATION OR A HEARING, OR TEMPORARILY DEFERRED, CAN GET ANY FORM OF GOVERNMENT SUBSIDY, ENTITLEMENT, OR BENEFIT FROM THE US GOVERNMENT.
This change will, with what Obama called “a stroke of the Presidential pen”, will require that ALL persons who lose their access to government funds be taken off the rolls. At that point, CMS should be required to work with DOGE to identify the persons affected, and remove them immediately, and without waiting for any court case to play out.
That will put a crimp into efforts to have the Federal government shoulder the cost of the many ineligible aliens’ use of funds designed for Americans.
The same with educational/college funds - the Executive Office will direct the Department of Education to identify all students receiving Pell Grants, student loan subsidies, and other educational handouts, and remove them from those programs.
Start doing that, and you will find that the self-deportation will begin - and, rapidly increase.
If Trump wants to appear magnanimous, he can direct that spouses and wives of currently serving military may be exempt for the duration of that spouse’s service.