The Europeans are doing a lot of exactly what you are describing (i.e. owning and operating N reg aircraft & occasionally offering training/renewals in them), mainly because of the onerous European licensing and maintenance regulations which CASA appear to be copying.
The main restriction in Europe is the right to work in the EU rather than any FAA restrictions, so I would assume the same would apply here. The only FAA restriction I know of is that ab-initio & certain other flight training needs TSA approval but renewals do not.