So what would happen if you had someone expert in both Talmud Law and also Constitutional Law?

That is we know the American Constitution is valid from the basic standpoint of: (1) a contract which is binding. (2)  The law of the country is the law. [Bava Batra ch 3. This is brought in the Rambam as being applicable much more than you would expect.](3) It embodies natural law as understood by the Rambam {Maimonides} and Saadia Gaon.

So what would happen if you had someone expert in both Talmud Law and also Constitutional Law? What kind of perspective would this bring? [ Probably an emphasis on traditional values and private property and limited power of the state]
In yeshiva I always considered Talmud law to the same thing as objective morality. And the civil laws of the state I thought of as more or less irrelevant.  I am pretty sure this was the general attitude. And I still consider that Talmud law to be basically Revelation--but not exactly. After all it is Revelation combined with human reasoning trying to figure out how keep the written law.
Still the whole concept of a State and its laws is morally relevant even on the personal level. Richard Epstein makes a very strong case against libertarians in regard to the State.

[A lot of Israeli judges have the Talmud in their private chambers.]

One thing about this is false  and demonic teachers of Talmud sprout up all the time like mushrooms after a rain. Unless you can really tell the difference between the real authentic holy teachers of Torah from Litvak places like Ponovitch or Brisk, and the Sitra Achra/Dark Side teachers, it is just not worth it to get involved. The religious world is basically mentally ill and therefore choose their leaders according to the higher degree of mental illness.