I do not think the mantel test is appropriate here, given your complex design. permanova may be better; you would not be asking "does my distance matrix look different when I use a different denoising/clustering method", but rather "does my distance matrix partition based on denoising/clustering method" which is probably a more straightforward and appropriate answer to your overall question: does denoising impact beta diversity.
The N=1 per site per time point does complicate things, but effectively site becomes a measure of "individual", e.g., in a multi-way permanova. If your question is "are these sites different?", you are sunk (unless if you have multiple time points per site). But since your question is "are these methods different?" You can run a multiway permanova with the formula site + week + method (or drop the week term if there is not significant temporal variation) for the purposes of benchmarking.
As @jwdebelius pointed out, there is no method that will allow you to determine whether individual samples are significantly different, unless if you sample them multiple times. You can aggregate your temporal samples (or bin them into groups of sequential weeks) to test for differences between site. But you cannot say "site X and week Y" is significantly different from other sites/weeks if you have N=1 at that site/week. Unless if temporal variation is high, I recommend aggregating (drop the week term)/binning to test for site-specific differences.