The EASA part FCLs state that a third country type rating may be put on an EASA licence limited to aircraft registered to the third country. To put the type on the ERASA licence a Licence Skills Test (LST) on that type must be conducted. As the only difference between that an a Licence Proficiency Check (LPC) is a raw data flown approach, it can be done as part of the induction LPS for the new company.
However, it depends on which EASA member state authority you deal with. For example, Ireland's IAA have no problems with putting the third country rating on your licence, but the UK CAA have not yet got a clear plan as how to do this (no surprises here!). I have been asking for quite some time how they intend to implement it and have had no answers yet. It seems they are telling pilots to do a LST on an EASA sim with an EASA examiner.
Might be a good time to change your licence issuing agency to the IAA.