Firstly, sorry for the extremely slow response...

Are you sure that it is only unresponsive during those times and no others?

Because of the CPU throttling that Amazon do on Micro servers, anytime that there is significant CPU activity for more than a fraction of a second the CPU will be throttled. Once the CPU is being throttled (almost back to zero in my experience) anything running will take up 100% CPU. While your appliance is using 100% CPU the throttling will continue (in a vicious circle) and even the most minor and arbitrary tasks will take hours and essentially make your server unusable. It will not start responding again until your

There are a few ideas on that thread you link to which may (but probably not 100% reliably) help you work around your issue. I guess you could try stopping backups altogether even? That might help if your site doesn't get many visitors and/or serves static html content, although TBH I wouldn't guarantee it... The harsh reality is that Micro servers are not really much good for anything real world production (private personal sites may be ok). There may be exceptions to that but it has been my experience...

So, unfortunately I suspect that the only real solution is to use a bigger server... Sorry probably not what you really wanted to hear I guess...