If the issue is what you linked to, which seems very similar to your situation, I'd try another module that supports SSL. If all you're doing is basically grabbing one HTML page, Mechanize is a bit over-featured for that, and you should have several other modules available to you as options. You could easily test and move to another module without many changes to your code.

If it's not that issue you linked to, I'd need more information on what's happening to diagnose it any further. But a longer timeout wouldn't actually "fix" it, as the issue would still remain. And for a small html page, you should be pulling that in in a second or two. So if you find yourself "solving" it by increasing the timeout, the problem will still be there.