Huge change in plans

There is absolutely no restriction in Hungary.
I entered the country few days ago with the US passport and during the highest alert last year as well.

The land controls do not exist - and if they do, here is the paperwork to download:

http://www.police.hu/sites/default/files/Igazol%C3%A1s%20%C3%BCzleti%20vagy%20gazdas%C3%A1gi%20c%C3%A9l%C3%BA%20beutaz%C3%A1shoz%20-%20Certificate%20for%20business%20travellers_0.pdf

You should download it, sign it, write there that adults in your group doing some computer business.
For “Tax Identification Number” use your SSN, the border police does not care less.
If you feel unsure, let your HR sign it and use the tax ID of the company.
Do NOT overcomplicate, just make sure all of you are vaccinated and - just in case - have a PCR test if you start your trip in Hungary.

BTW: I also have a Hungarian passport and tested “our” border guards extensively.
Few days ago entered Hungary from Croatia with my car having US license plate.
No question was asked, not even above paperwork or vaccination. I have a French vaccination certificate… never had to show it!

The best way is if you download that paperwork, fill it out and call the Hungarian consulate.
Just tell them that you have the Hungarian “entry for business or economic purpose in pursuance of title 5/A. of Government Decree No. 408 of 2020 (VIII. 30.)” filled and verify, that there won’t be any problem.

Airports are somewhat more risky than the land borders though. My kid recently came from Göteborg by airplane. He had the same paperwork. The first three passengers of the airplane were asked some COVID-related questions, than the border guards disappeared.
(Göteborg is in Sweden, there are Schengen-rules). My kid wasn’t among the first, he was even disappointed that he had to ask his internship place(!) to fill out this paperwork.

Don’t be scared, this is a complete farce.