First of all, welcome to LRH!
Theres a couple of options other than shimming those bases, but ill cover shims first. The first thing id do if Iwere in your position would be to find out how much adjustment you have above your current zero. That way we will know how much adjustment you have in both directions, and you can optimize the location of your zero, inside the scope, so that your zero is towards the bottom of the adjustment, leaving you plenty of "up" adjustment.
at the bottom of this picture you will find the formula for computing shim thickness needed for a given moa of adjustment wanted.
The problem is that there is a limit to the amount of shimming you can do without damaging the scope. usually about .020 on a long action rifle. in this instance about 12moa, or just over 24" at 200yds. You can buy shims made of steel from any of the mailorder catalogs, or your local gunsmith will probably be willing to sell you a few, but its easier to make some yourself from a spare aluminum sodapop can you have around. It measures .004" thick. If you do decide to shim, don't be like one of the guys I saw at the range a while back. Put those shims under the rear base, not inside the ring [img]/ubbthreads/images/graemlins/grin.gif[/img]!!!
My suggestion would be to look at getting either a tapered base, or a set of the Burris Signature rings with interchangable inserts. Thats what all of the top portion of the above photo is. It gives all of the possible shim values with these rings, and these rings combined with a variety of tapered bases. Any of the tapered bases for the remington will fit the howa as you already mentioned.
I believe that leupold makes a 20moa base that will fit your rings, and if I recall its only in the area of about 30 bucks.... worth a look.