UK Car in DE

I'm going to be working in Lux City for a year+ and I feel while I get settled I'll take up residence in either LU, FR, BE or DE. With the current pandemic the only real way for me to get there from UK (where I am now) is the drive with my basic stuff in the boot.

I am reasonably sure I can keep my car for six months on UK plates then I may well move to Lux City proper with no car (take back to UK or even keep it in Lux on UK plates for another six months).

Anyway, from my research it seems taxing a car on in BE is very expensive, so that's out, so I'm posting this in LU, FR and DE sub-fora so I don't make an expensive cocked-up decision. I'm not bothered where I'd stay but obviously somewhere nice is preferable.

I love my car, it's a proper GT for touring, a Maserati Quattroporte, 2006, 4,2 litre, horsepower is 405hp. Value is £9k in UK, prices dropping like a stone.

So I'm looking for rough costs of road tax, any import or registration and insurance if possible, I'm a 58 yo male been driving with no convictions at all for 35+ years. I can drive on my UK insurance for 90 days per trip so I could maybe get round it by going home and coming back, though that feels slightly immoral and risky to me. I will still have my house in UK and what's lets of my family living there!

Alternatives are keeping it on UK plates as long as possible and taking it back to UK over winter say, or moving about to get six months in various nearby States and go from there....

Or just not bother and lay it up here. But I do enjoy it, it's a hobby and I'd miss it.

I will use car for local shopping etc, and touring at weekends, not for commuting, I'm not mad....

You might be under-estimating the size of Luxembourg. I doubt that staying in any nearby German city (e.g. Trier or Bitburg) and commuting (by public transport) to Luxembourg city is feasible. The other countries you mention look even further on the map.
In any case, your right to stay in any of these depend on arrangements post-Brexit, which are still entirely unclear. (Until end of this year, the pre-Brexit rules apply.)