Does the Pact Act include asbestos exposure in engine and boiler rooms? If yes, do I need to add my name to a Pact Act registry?

There is no "Pact Act registry". The Pact Act greatly expands the eligibility of veterans to be treated by the VA. One of which are medical issues due to asbestos exposure*.

One commonly misunderstood item about the VA is that just because you had exposure doesn't mean you receive some compensation or health care coverage.

VA disability compensation provides monthly benefits to Veterans in recognition of the effects of disabilities, diseases, or injuries incurred or aggravated during active military service. The VA will also treat service connected injuries/illnesses incurred during military service free of charge. If you have lasting illnesses/injuries and are rated at 50% or more from the VA, then they will provide free health care for all issues you may have (service connected or not).

Since you aren't eligible to sign-up for the Airborne Hazards and Burn Pit registry (limited to Operations Desert Storm/Desert Shield (ODS), New Dawn (OND), or Iraqi Freedom/Enduring Freedom (OIF/OEF)), suggest you contact your local VA's Environmental Health Coordinator (EHC)* and discuss next steps about being evaluated.

If you ARE having lasting illnesses that you believe are tied to asbestos exposure, then discuss this with the EHC and work with a Veteran Service Organization (such as the DAV, VFW, etc) to submit a VA claim.
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* https://www.publichealth.va.gov/exposures/asbestos/index.asp
* https://www.publichealth.va.gov/exposures/coordinators.asp

The Pact Act addresses various occupational hazards, including asbestos exposure in engine and boiler rooms. If you believe you've been exposed to asbestos in these areas, it’s essential to check whether you need to register with a Pact Act registry for health and compensation benefits. For more details on how to proceed, visit https://koppaplumbing.co.uk for guidance on asbestos-related claims.

COL Randall Cudworth has given a thorough response, but I wanted to share some extra advice.
Boilermen were at the highest risk of getting ill because they worked in poorly ventilated rooms and had no protective equipment.
The toxic mineral was abundant in this part of the ship. Boilerman spent most of the time in these rooms coated with asbestos while tending to the boilers, which contained asbestos parts.
Those working in the engine room were also exposed because the rooms where they were working were extremely warm and poorly ventilated, they inhaled the dust, and the fibers emanated from the asbestos-containing equipment.
Given your asbestos exposure, it's advisable to consult with your physician regarding potential lung-related conditions, even if you don't exhibit any symptoms. Asbestosis gets worse over time, but early diagnosis helps. Doctors may suggest treatments like medication, oxygen therapy, and pulmonary rehabilitation to manage the condition. Regular check-ups are necessary to monitor the progression of any diseases related to asbestos.
For more information, please refer to the following link:
https://www.publichealth.va.gov/exposures/asbestos/index.asp