The Gohmert suit is as silly as the Plaintiff's name would indicate. Leaving aside the obvious questions about standing, whether there is an actual "case or controversy", etc., the argument in the Complaint is that there is a conflict between the Constitution and the Electoral Count Act concerning the method of resolving objections. Specifically, the Complaint asserts that the ECA "limits or eliminates [Pence's] exclusive authority and sole discretion under the Twelfth Amendment to determine which slates of electors for a State, or neither, may be counted; and replaces the Twelfth Amendment’s dispute resolution procedure – under which the House of Representatives has sole authority to choose the President." In other words, Gohmert is arguing that Pence, as President of the Senate has almost unbridled authority to conduct the voting process on Jan. 6, and by creating a structure for resolving disputes, the ECA impinges on that authority. The basic premise of that argument is that Pence does in fact have such authority.
All the 12th Amendment says is: "The President of the Senate shall, in the presence of the Senate and House of Representatives, open all the certificates [of the elector's votes] and the votes shall then be counted." This is a perfunctory duty - open the certificates and count the votes. There is not one word in the Constitution about objections to electors, how the objections must be lodged, or how objections are to be resolved. No additional authority is given to the VP to do anything but open the certificates and count the votes.
By arguing that the ECA resolution dispute process is unconstitutional, there must be an assumption that objections are allowed in the first place. The Constitution does not provide for objections; that right is only granted by the ECA itself. The internal inconsistency in the logic is staggering. Gohmert, et al are arguing that Congress has the power to create a mechanism for lodging objections (not found in the Constitution) but not the power to set up a process for resolving the objections. The VP's duty to open the certificates and count the votes is being construed as absolute authority to do whatever he pleases. So much for strict construction. If the ECA is unconstitutional, then arguable there is no right to object at all.
Only the electors selected by the various states, in the manner determined by their legislatures, can have their votes opened and counted. Those electors are the ones certified by the states - not the ones determined by Pence.