Puffin is cheap, so it's probably the first of the two you should try, if you don't get a better opinion. Whether either of them will work well, if they work at all, with your sites is problematic. If you can post the site's URL (and it is accessible to guests) other's might be willing to give it a try and tell you their results.
I noticed that Puffin was updated a few days ago, and that brings up an important point. Check when each app has received their last updates. If it hasn't seen an update in over six months it may no longer be supported. For some app types that's not terribly important, but browsers need maintenance updates.
None of the Flash enabled browsers are great. They tend to be slow, and the UI elements don't always translate nicely. The only way to know if they will work is to try them.
If your sites are actually using Java and not Flash, you might want to give Opera Mini a try. It's free.
As always, make sure to search the App Store and see if your site has an app. Many of the popular sites do, and often they work better than the site.
