A big problem is that you only have 1 observation for each of tool4 and tool5, those were both used only under cause3, with tool5 only used on side1 and tool4 on side2. Also, tool1 only seems to have been used on side2. And cause1,side1 only was handled once, with tool6. That limits your ability to disentangle those tools from those particular sets of conditions.
Without more information about the nature of the study and what led to the major differences in the various tool values, it's hard to see how to proceed reliably. If the outcome variable is itself the choice of tool under combinations of side and cause--that is, the outcome is the set of numbers within the cells of your table--there might be some solution, but the small number of overall cases (fewer than 40, by a quick count) indicates that you will have trouble in any event.
Added in response to edited question
The problem with proceeding as you propose:
first analysis to see the effect of cause, if not significant then I am OK with pooling at least two of the causes (2 and 3) so then I will have two main cause groups. Then run another analysis within the collapsed cause group for the effect of side and tool/prosthesis.
is that you are using the results of the study to determine how to analyze the study. That's not generally good practice. At best, the p values that you get at later steps will be unreliable as they are based on an assumption that tests were pre-specified without seeing the results. See for example Chapter 4 of Frank Harrell's course notes for guidance on multiple-variable modeling.
It's best to think through carefully just what hypotheses you can reasonably hope to test with fewer than 40 observations. In a biomedical study that's usually only enough to handle about 3 predictors when there's a continuous outcome (10-20 cases per predictor). Your cause with 3 levels already counts as 2 predictors; side by itself is another, and if you include interactions with cause that's 2 more. Even as a random effect, tool is at least 1 beyond that.
So: Do you care more about the association of function with side or with cause or with tool? Does your knowledge of the subject matter suggest that 2 types of cause would have similar effects, or that you might reasonably combine some types of tool? Maybe you could even just combine all tool types together at this stage of your study?
An alternative to avoid overfitting this limited data set would be to use a penalized approach like ridge regression, particularly if you could combine data across some or all the levels of tool. The imbalance in the data set could still pose difficulties for interpretation, however.
