The construction would be a private company that works with the government. That exists already - I had applied for one before - and the pay was completely competitive, for German salaries. 80K.
This might be possible with public grants, but I've never seen it. They usually come with additional strings attached, such as being a de minimis subsidy (limited to 300k in a rolling 3 year window), requiring private co-funding (e.g. 50% for the whole project), being limited to a couple of months or maybe 2-3 years max, etc., so for competitive pay, you must almost always rely on some kind of "mixed financing": every penny above the salary that TVöD allows must be acquired from sources other than the gran. ymmv, part of the lack of attractiveness of public funding is the fact that each program differs in a million different ways.