H1B cap exempt

Hello,

I am considering applying for an H1B visa but I am not sure if my best bet is to apply as a cap-exempt. I work for a non-profit that has an AGREEMENT with a university to offer a certificate and/or masters in non-profit management at a reduced cost in the agency's facilities. Since this is not an affiliation I am not sure if it will make a strong enough case for USCIS.
Does anybody have experience in this matter?

I would also want to know how long does it usually take to get the Certification from the DOL that your employer files for you and how long it takes to submit all the immigration forms.

All information and advice is very appreciated!