You’re right that your watch is not going to give you an accurate measure of training stress for strength workouts. In fact, since you’re using TP and looking to use the TSS/CTL measure as an overarching guideline, I would recommend not even wearing your watch for strength sessions (or not recording the session) and instead log them manually in TP with an estimate for TSS.
Here are some articles and forum threads around this topic (more or less specifically) that you might find useful in understanding how to score your strength sessions.
Understanding and Using the TrainingPeaks Metrics CTL and TSS
Recommendation for logging miscellaneous training in Training Peaks
Does that help?