It is definitely useful to log a TSS score for all your workouts if you want to have a reasonably accurate picture of your body’s state regarding readiness for a hard effort and fatigue (which have an inverse relationship: lower fatigue = higher readiness, and vice versa).
For things that aren’t accurately measurable with a HR monitor, such as strength training and climbing, you’ll need to estimate a TSS. A good way to do this is to review the hrTSS (make sure you click the down arrow to change TSS or rTSS to hrTSS if you need to, when you’ve used the HR monitor) for a number of runs and hikes to get an idea of what effort equates to what approx. TSS. For instance, for a full out Chamonix Mountain Fit session, the UA coaches I’ve worked with have used 50 as the TSS for that workout. For an hour of gym-based strength with weights, you’re probably going to use a higher number.
It’s obviously not as cut and dried and scientific as using strap + monitor for aerobic efforts, but it’s a lot better than not including a TSS for those other workouts at all. Over time you’ll get a good sense of your TSS-effort equivalents. Remember to always base it on how effortful it felt *on that day*. Some days you’re going to have tons of energy and strength, and other days the same workout will seem much harder. Ideally you would reflect that in the TSS score.
However, that all being said, don’t get too hung up on the precision of the numbers. The main thing is to have a fairly accurate picture of the stress load that you can use to track the effect of your training over time.
Here’s another forum thread on this topic: https://uphillathlete.com/forums/topic/climbing-and-trainingpeaks/
This article also gives guidance about modifying the hrTSS when you’re carrying weight, doing a lot of vert. etc. on aerobic outings.