There’s no longer “large OSV training programs”, you need time over 3,000 GT (and as chief mate) to get it.
You should be able to get a tonnage limit of 5,000 tons for service on a boat just under 3,000 GT under paragraph c.
From 46 CFR 11.493:
(b) The minimum service required to qualify an applicant for master (OSV) of 1,600 GRT/3,000 GT or more is 24 months of total service as mate, chief mate, or master of ocean or near-coastal and/or Great Lakes on self-propelled vessels of more than 100 GRT. At least one-half of the required experience must be served as chief mate and be obtained on vessels of 1,600 GRT/3,000 GT or more.
c) If an applicant for master (OSV) of more than 1,600 GRT/3,000 GT does not have the service on vessels of 1,600 GRT/3,000 GT or more as required by paragraph (b) of this section, a tonnage limitation will be placed on the officer endorsement based on the applicant’s qualifying experience. The endorsement will be limited to the maximum tonnage on which at least 25 percent of the required experience was obtained, or 150 percent of the maximum tonnage on which at least 50 percent of the service was obtained, whichever is higher. However, the minimum tonnage limitation calculated according to this paragraph will be 2,000 GRT. Limitations are in multiples of 1,000 GRT using the next higher figure when an intermediate tonnage is calculated. In no case will the limitation exceed 10,000 GRT/GT for OSVs unless the applicant meets the full requirements for an unlimited tonnage endorsement.