Does the house have the ultimate say in this if there is a dispute? If it does, if I were the Trump campaign I’d ask for hand counted recounts in the mail in states, drag the process out until January and have the House decided the matter.
No. The Electoral College is purposely a stand-alone temporary body made up of people who hold no other political office. Hamilton wrote about this in Federalist #68:
Nothing was more to be desired than that every practicable obstacle should be opposed to cabal, intrigue, and corruption. These most deadly adversaries of republican government might naturally have been expected to make their approaches from more than one querter, but chiefly from the desire in foreign powers to gain an improper ascendant in our councils. How could they better gratify this, than by raising a creature of their own to the chief magistracy of the Union? But the convention have guarded against all danger of this sort, with the most provident and judicious attention. They have not made the appointment of the President to depend on any preexisting bodies of men, who might be tampered with beforehand to prostitute their votes; but they have referred it in the first instance to an immediate act of the people of America, to be exerted in the choice of persons for the temporary and sole purpose of making the appointment. And they have excluded from eligibility to this trust, all those who from situation might be suspected of too great devotion to the President in office. No senator, representative, or other person holding a place of trust or profit under the United States, can be of the numbers of the electors. Thus without corrupting the body of the people, the immediate agents in the election will at least enter upon the task free from any sinister bias. Their transient existence, and their detached situation, already taken notice of, afford a satisfactory prospect of their continuing so, to the conclusion of it. The business of corruption, when it is to embrace so considerable a number of men, requires time as well as means. Nor would it be found easy suddenly to embark them, dispersed as they would be over thirteen States, in any combinations founded upon motives, which though they could not properly be denominated corrupt, might yet be of a nature to mislead them from their duty.
Federalist #68 says that the Electoral College was meant to be filled temporarily by citizens of the United States who are free from other obligations to federal office holders (such as Congress overseeing what they do). Furthermore, they were not to convene as a single body where factions might form; they were to remain local to their states and vote separately from the other states.
Note the phrase "...an immediate act of the people of America, to be exerted in the choice of persons for the temporary and sole purpose of making the appointment."
The electors were chosen to "make the appointment" of the President, not the legislatures nor the Congress. Any disputes within a state must be resolved by the Electoral College that meets in that state. Either the state legislature appoints its Electors directly to resolve a dispute, or the College members meet to hash it out (unless there are "faithless elector" laws in the state).
The only role for Congress is to break ties in the Electoral College.
-PJ