Simply make the bar 100% height.
If you have 12 icons you can just-about-get-away-with-though-I-shouldn't-recommend-it making them expand to fill the available screen space. 1/12th of a 4 inch screen (75% of current users have 4+ inch, and that includes somewhat older phones) gives you 7.33²mm squares.
Windows Phone recommends to use 7²mm squares but with 2mm whitespace to prevent accidental taps on neighboring buttons, so a total of 11²mm per button. However, their data shows that at smaller sizes, the amount of missed touches might still be acceptable. (via this post)
The obvious downside here is that you've got a higher chance at people mis-tapping,
The upside is that your site will feel pretty much the same on any device, and that all icons can be viewed at once.
Make the menu double-wide
Alternatively, make your menu two columns wide. Put the top-used 5 icons on the left, always visible, with a hamburger (plus "more" label) at the bottom. When tapping the hamburger, expand the menu in to an (almost-?) fullscreen table of 2x6 buttons with labels.
The downside here is that you lose a quarter of your screenspace (though you could narrow it down a bit) which is a LOT. And you're still hiding some functionality behind a second menu.
Consider an alternative layout
If you're going to work on mobile, you really don't have much width to play with. It's also not something you easily scroll away from. How about a bar at the top? If it's absolutely positioned, users will see the menu when they load a page, but as they scroll down it'll disappear and they have 100% of the screen to view content on.
Or use a slightly more elegant solution like this.
Of course this wouldn't be using a 12-wide bar, but something like a 4x3 grid. It'll take up a bit more space vertically, but you'd scroll past that pretty quickly.
Upside here is that you're still showing everything with decently tappable buttons, while still allowing for content to fill up the screen.
Downside is that on page load there's quite a face-full of buttons, and that they're not always directly accessible.