It looks like the ABPN allows part time fellowships. You would have to get a program to agree to it and allow you to moonlight.
I copied the info below from 2020 Information for Applications document from the ABPN website.
CAP:
The required two full-time equivalent (FTE) years of specialized training in child and adolescent psychiatry may be completed on a part-time basis, as long as it is not less than half time; credit is not given for periods of training lasting less than one year, except under special circumstances that must be approved by the ABPN Credentials Committee. In such cases, it is the responsibility of the applicant to provide detailed documentation from the respective training directors, including exact dates (month/day/year to month/day/year) outlining training content, duties, and responsibilities. Each case is considered on an individual basis. The two years of full-time, specific training in child and adolescent psychiatry may be taken in no more than two blocks. If completed in two blocks, the blocks must not be more than five years apart.
Forensics:
The required one full-time equivalent (FTE) year of specialized training in forensic psychiatry may be completed on a part-time basis, as long as it is not less than half time; credit is not given for periods of training lasting less than one year, except under special circumstances that must be approved by the ABPN Credentials Committee. In such cases, it is the responsibility of the applicant to provide detailed documentation from the respective training directors, including exact dates (month/day/year to month/day/year) outlining training content, duties, and responsibilities. Each case is considered on an individual basis.
