Unless you intend to fly a CJ4 (or maybe a M2) I wouldn't spend anything on the CJ1 conversion. The base model of the CJ is so similar (C525, CJ1, CJ1+, CJ2, CJ2+, CJ3, CJ3+) that on type and theoretical training with your future operator should do (my opinion). If prices are the same I would choose the course depending on the future avionic package that your CJ1 have.
If it's a basic CJ1 the avionics could be Proline 21 displays with Universal UNS-1 FMS, maybe paired with Garmin GPS - try Flight Safety in Farnborough. The use a classic CJ2 sim.
If it's a CJ1+ AAA in Austria is exactly a CJ1+ model. Starwings in Dortmund uses a similar cockpit (CJ3) and transition would be quite easy.
Also worth checking would be Atlas Air Service in Bremen.
https://www.aas.ag/flugbetrieb/ato-pilotenausbildung
They used to train in the FNPT2 that is operated in cooperation with Lufthansa Flight Training. The mandatory training on type is performed with their own aircraft including the checkflight.
Good luck!