I am particularly concerned with whether or not the task load index could be
considered an interval variable,
This is a fundamental assumption people make when constructing, administering and analyzing any measure under the classical test theory (CTT) paradigm i.e. count items and add 'em up.
That is to say, your variable is certainly treated as though it is linear, but are the scores actually linear? is it valid to combine scores on each factor to derive another score?
Even further, conceptually, let us presume that two individuals have the same task load index. Do they possess, quantitatively, the same amount of the latent variable? Not necessarily.
This is a fundamental assumption and inherent flaw in classical test theory.
Can a difference between two means (e.g., 50 and 60) be interpreted as equal to a difference between two other means (e.g., 10 and 20)?
They could be, but this is an assumption you are making by applying classical test theory. As noted above, you are also assuming that two individuals with the same score are equal. This is also an assumption that may or may not be correct.
If not, what are the consequences of using a wrong test?
This is a no-brainer and I’m sure you know the answer to this: the wrong inferences will be made regarding results, and how they are applied in a meaningful way.
Suppose perceived workload is non-linear and instead, say, exponential, and it is tightly correlated with the "likelihood of totally spazzing out." NOTE: despite earnest lobbying on my behalf, I was unsuccessful in convincing the APA to consider Low Spaz Threshold Disorder for inclusion in the DSM-V. Oh well, there is always DSM-VI. notwithstanding, incorrectly assuming workload is linear might lead you to erroneously think that someone with a score of 70 has only a slightly higher likelihood of spazzing out relative to someone with a score of 60.
What is a suitable test?
If this actually concerns you - and it should if you care about validly measuring the construct, look into item-response theory (IRT) which departs from the CTT that is commonly applied.
Briefly, IRT is a statistical model that specifies the probability of each response option to an item as a function of the target trait thought to be measured. Respondents are no longer scored by the number of items they answer correctly. Instead, they are assigned an estimate of their location on the underlying distribution of ability (ϴ). Although it's commonly applied for exams, where the term ability is meaningful, ability can also be taken to mean level of depression, stress, happiness, procrastination or any other latent trait of interest. In your case, perceived workload.
IRT can take into account the non-linear relationship between an individual's score and their level of ϴ. When estimating a person's ϴ, parameters such as item difficulty and discrimination can be taken into account.
For each item, one can derive an item characteristic curve (ICC), which maps the probability of endorsing an item according to an individual's ability. Think of ϴ as being scored in standard deviations. This figure shows the ICC for a dichotomous variable.
Of greater relevance to your problem, this can also be done for items that are polytomous. Note how the estimated ϴ shifts in a non-linear fashion as we go from endorsing 0, 1 ... 3 in step-wise manner. That is, within a single item, the level of ϴ can differ substantially between likelihood of endorsing "agree" versus "strongly agree."
A test characteristic curve can also be derived as shown below. The true score is the essentially the raw score. As you can see, about 15 points separate people with ϴ = -1 vs. 0, while about 10 points separates ϴ = 1 vs. 2. Once ability estimates are determined, appropriate statistical tests can be applied.
In theory, I think you could apply IRT to the NASA-TLX. You would probably need to treat each subscale as though it was actually an item, with the workload index as the summed score. There are also several assumptions that need to be considered when applying IRT. One important one in your context is that it assumes that single latent trait is being measured. As I know little about the NASA-TLX, it's unclear to me if this is the case. And you will need a large dataset to calibrate the curve functions.
A simple but informative primer can be found here:
A Simple Guide to the Item Response Theory (IRT) and Rasch Modeling
Note: the Rasch model is the most common model of IRT, but it deals with dichotomous data. What you're after is a polytomous model, but you would do well to first understand how the Rasch model is applied.