Do You Need A Security Clearance To Work In The Training Room?

There is no requirements to have a security clearance to handle PII (Personally Identifiable Information). It's not classified information.

PII in the military is falls under SBU or CUI - Sensitive but Unclassified or Controlled Unclassified Information and is covered under the Army's Privacy Program* and not the Army's security regulations (i.e., AR 380-5 and others in the series).

There IS a requirement for privacy act training if you are handling PII and penalties for mishandling it - reference the program for details.
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* The Army Privacy Program - https://www.rmda.army.mil/privacy/RMDA-PO-Programs-Privacy-Program.html

COL Randall C. nailed it.

If a Soldier In a clearance requiring MOS or shop loses their clearance - or has access suspended pending investigation - training room is one of the few places a Soldier can go and still be a productive member of the team. Training room, supply room, motor pool. That's pretty much it. Saw it more than a couple times working in MI.

You may be confusing security investigations with security clearance because the various Investigation levels can qualify you for a clearance if you have a “need to know”. Presumably, he accesses DTMS so he should at minimum have a favorable background investigation or tier 1 (T1) or equivalent investigation. A favorable T1 investigation can be used for a confidential clearance if the position requires classified access. T3 qualifies for Secret and T5 for Top Secret if there is a requirement.