When your goals is to measure one option over the other
As part of a Human-Computer Interaction (HCI) study I also needed to measure cognitive load in order to assess whether a particular user interface could reduce perceived cognitive load (as compared to a control group). Because of practical reasons (and not to intrude too much in 'ordinary' computer usage) I decided to stick to the commonly used NASA TLX scale (Hart, 2006). However, for small-scale studies (mine involved just 16 participants) I have the impression you are just as well off merely asking for qualitative feedback. Of course you can't run statistics on these, but interpreting effect size on such small sample sizes is arguably meaningless either way. You could thus simply ask participants which option is preferred in regards to perceived cognitive load and report on this as a 'significant' result. Of course I can not guarantee whether reviewers would agree. :) In the case you have suitable existing NASA TLX measures to compare against this still remains a worthwhile option.
However, as you state a big downside to such a scale-based test is that it can only be administered at a few points throughout the study, since filling out scales is time intensive.
For large scale or continuous quantitative studies
Skin conductance, blood volume, and pulse rate seem quite promising and are being used in HCI (Janssen et al., 2015; Mark et al., 2012), which are indicative of arousal. These nowadays can more easily be measured in an easy form factor (e.g., a watch). However, it is important to mention that arousal is not indicative of valence—the
pleasantness or unpleasantness of stimuli. Neither is using such equipment straightforward. Establishing a baseline for participants takes a considerable amount of work and care needs to be taken to do it properly (if one wants to compare results with other participants). Measurements on the wrist (as in measured by a watch) are weaker than on palms. For some people, no suitable signal is measured at all. Environmental factors like temperature changes and movement (in regards to skin conductance) and changes in lighting (pupil dilation) need to be controlled for.
To answer one of your containing questions, I do not believe there is an established opinion or a preferred method and it still relies heavily on what can be measured in the given context and what is to be expected (e.g., can it be presumed there is no positive valence?). Depending on the study one could still be preferred over the other (does the task involve movement?), and I definitely would not rule out qualitative measures.
Studies comparing different measures
When I was looking for information I did run into some sources which compare different measures. However, I can not summarize them by heart, but I hope the current mentioned issues and sources can already guide you in the right direction.
Hart, S. G. (2006, October). NASA-task load index (NASA-TLX); 20 years later. In Proceedings of the human factors and ergonomics society annual meeting (Vol. 50, No. 9, pp. 904-908). Sage Publications.
Janssen, C. P., Gould, S. J., Li, S. Y., Brumby, D. P., & Cox, A. L. (2015). Integrating knowledge of multitasking and Interruptions across different Perspectives and research methods. International Journal of Human-Computer Studies, 79, 1-5.
Mark, G., Voida, S., & Cardello, A. (2012, May). A pace not dictated by electrons: an empirical study of work without email. In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (pp. 555-564). ACM.
Forne, M. (2012). Physiology as a Tool for UX and Usability Testing. School of Computer Science and Communication, Master. Royal Institute of Technology, Stockholm.
Haapalainen, E., Kim, S., Forlizzi, J. F., & Dey, A. K. (2010, September). Psycho-physiological measures for assessing cognitive load. In Proceedings of the 12th ACM international conference on Ubiquitous computing (pp. 301-310). ACM.