Kruskal is fine if your ANOVA is one-way, and it takes non-normality into account, as does PERMANOVA. You just create one big table with all the raw data and perform the PERMANOVA on that. Do not summarise data for analysis. Then whatever R2 and p values you obtain, you need to explain them, rather than manipulating the data to obtain your preferred result, which is unscientific :)

Although not specifically related to your question, one issue you need to take into account when performing PERMANOVA is multivariate dispersion (equivalent to homogeneity of variance in ANOVA). You need to test whether the dispersion of observations the observations among groups (years, seasons, etc) is the same, otherwise the results from PERMANOVA won't be valid. Check the documentation for the function in vegan. There is another function in vegan, betadisper, that allows you to perform the test.

Now, doing a multivariate (eg, PERMANOVA) or a univariate (KW, ANOVA) analysis has nothing to do with the issue you mention, but to the hypothesis and research question you have, as the approach for either alternative answers a rather different question.

As for 'when I summarize the data', perhaps you need to provide more details on what you are actually doing to better understand if it makes sense. However, in most case, data are data and manipulating it to obtain a desired result may not be the best approach, as already noted.

Kruskal ANOVA is univariate and would ignore the multivariate nature of the question. It is also true as Jaime Pinzon says that applying a test that assumes homogeneity of dispersions may not be appropriate if there is a presumption that different fish have different breadths of diet (specialists vs generalists, for example).

You are getting small differences that are highly significant because the numerous samples give you massive power, but you are in danger of missing the bigger picture. The underlying problem here is the sample grain, and this is well understood for such dietary studies. What is found in an individual fish's stomach only tells you what it has just eaten. The data are dominated by individual-individual noise. To improve the 'signal (diet) to noise (last meal)' ratio it is often appropriate to redefine your samples as the sum or average of a number of fish guts. Choose a number (say 4, or 5, or 10) and then randomly sample groups of fish of that size from your data within each of your combinations of factors. This will give you much more robust data relevant to differences in diet among levels of factors, rather than differences in last meals. Finally, the fish are 'doing the sampling' so you have no control over the sampling effort, so it would be sensible to standardize (convert to percentages) and maybe do a mild (square root) transformation prior to calculating appropriate resemblances (e.g. Bray-Curtis).

For hypothesis testing, if you aren't too bothered by differences in dispersion Permanova may be OK. Alternatively multiway ANOSIM (which gives measures of relative effect sizes) would be an appropriate method, although it's not in R yet as far as I know. For more see Lek E., Fairclough D. V., Platell M. E., Clarke K. R., Tweedley J. R. & Potter I. C. (2011) To what extent are the dietary compositions of three abundant, co-occurring labrid species different and related to latitude, habitat, body size and season? J. Fish Biol. 78, 1913–1943 for a fish example (they used a sample size of 4 fish I think). For comparison of Permanova and Anosim see doi:10.1111/aec.13059. 3-way ANOSIM (incorporating the Lek et al. data) coming soon in the same issue.