There's more to the EP clause in terms of how a court would look at it. The level of scrutinyapplied by the court is based on whether the disadvantaged class of people is protected. Such protected classes typically involve race/alienage (strict scrutiny - whether the government act was necessary and narrowly tailored to serve a compelling interest) or sex (intermediate scrutiny - whether the act furthers an important government interest through substantially related means).

Unfortunately, political affiliation is not a traditionally recognized protected class, and therefore rational basis review would likely be the standard in a Equal Protection challenge, i.e., the act was rationally related to a legitimate government interest. The government actor almost always wins under this standard.

IMO, they need a better basis for a claim than the equal protections clause. I think most of this will rest on procedural challenges that appear to have strong legs. I'm sure the legal minds on team Trump will make every case they possibly can.