I see two main reasons for using VPS (though there are many other, lesser, reasons):
1. The site owner wants a guaranteed share of resources and/or immunity from the actions of another site owner on a shared server which may bring the whole server down.
2. The site itself has outgrown shared hosting because of the demands on resources it makes. This would appear to be your situation. 1500 SQL queries per second is incredible, and definitely shows something wrong with the scripting or settings of your website. Even with 80k products and an active site it shouldn't generate that number of SQL queries (per minute maybe, but not per second). We have a few customers with 120k+ product websites and they don't place a strain on our shared servers - which again points to a problem with the scripting or settings used on your site.
And having said that, given the demands of your site, having a VPS with just 256 Mbs of RAM is a bad joke. 256 Mbs is the bottom rung of the VPS ladder and I would only recommend that for a not-too-active site which fell under the reasons I gave in Para 1.
Okay, I'll shut up about VPS now. Your problem is that your site works up until you try to use a particular template - at which point it stops working. You've said that the same site with the Classic Green template works okay.
I've spent a lot of time chasing down problems like this and point again to the frequent use of PHP Short Tags by developers of templates. I'm as sure as I can be that this is where you'll find the problem.
You can test this out by adding this to a root level .htaccess file:
php_flag short_open_tag on
Depending on the configuration of the server this will either work and allow the use of short tags, or else it will cause a "500 - Internal Server" error.
php_value short_open_tag on