I have 4 months remaining in my contract. Unit isn’t being cooperative as far as letting me start the SFL-TAP process. What should I do?

Yes HOWEVER, if done in an adversarial mode meeting the letter of the reg, vice perhaps the intention..
a SM can meet meet Career Readiness Standards (CRS) in about 9 days.

Complete the Individual Transition Plan (ITP) documenting the individual's personal employment, higher education, career technical training, and/or entrepreneurship goals, actions and milestones.
Complete an integrated 12-month post-separation budget.
Register on the VA eBenefits site.
Complete Continuum of Military Service Opportunity Counseling (Active Component to Reserve Component only).
Evaluate transferability of military skills to the civilian workforce and complete the gap analysis provided during the MOC Crosswalk.
Identify requirements and eligibility for certification, licensure, and apprenticeship in the Service member's desired potential career field.
Complete a standardized individual assessment tool to identify personal interests and leanings that will enable informed decision-making regarding career selection.
Receive a Department of Labor (DOL) Gold Card for American Job Centers. The Gold Card provides six months of priority services for post 9/11 Veterans at any DOL American Job Center, including DOL-funded employment and training programs; case management; skills assessment and interest surveys; career guidance; and job search assistance.
Complete job application package or receive job offer letter. The package must include a private or public sector resume, personal and professional references, and at least two submitted job applications.

That is NOT what was intended or desired for our transitioning SM's

So slapping AR 600-81 on the desk and saying "See Sir, Chap 4 says you MUST allow me to do this stuff" is a fast way to be escorted through the process by a tasked pissed off SGT who would rather be training his troops......not escorting someone on ETS status that wants to shout about regs....

Perhaps , using AR 600-81 and the local TAP assistance available...
Make a schedule of activities you must attend/complete by reg and the ones you would like to attend, classes or briefings you want to sit in on, time you wish to allot using resources, meeting with mentors or TAP assistance folks.

If a desired briefing is offered on Tues and Fridays ...and you have CQ on Tues... Dont schedule your briefing to interfere with a known tasking.

If the company annual APFT is Monday next week, don't go find a briefing to sit in on for that day, when it is offered EVERY day or every week.

If your schedule is 50% or better not interfering with known training (Dont forget you still wear a US Army Name tape.. "Uncle Sam Aint Released Me Yet").. you are subject to the company training schedule until the day you depart on leave.et"), tasks, or other planed and known events you are reasonable expected to participate in.

If you have a resume workshop Tuesday morning and a company 100% urinalysis is called that morning..... and you are allowed to show up any time of the day for that workshop.... Dont throw the "TAP" card... give the sample and then go to class.

Transition 4 months out with an SM willing to be an adult about it and work within the needs of the unit...will get a lot of support.
Kicking and screaming IG IG IG..will get you the bear minimum required by the regulation
in the most inhospitable manner possible.

NOTE: MSG (Join to see) Did not, nor from reading his many responses here advocate being "that Guy". I took his post as one of being educated in the process before you go speak to leadership. My response here was less to his and more in general of a way to approach the issue successfully winning the war, vice just the battle.