The Adcom 5300 has an input impedance of 49,900 ohms. So, you need to look for a tube preamp with an output impedance of 4,900 ohms or less.
For the subwoofer connection, you'll need an rca y-adapter (2 female rca connectors feeding to one female rca connector). Run a single male-to-male rca cable from the subwoofer/LFE output on your processor to one of the female rca inputs on the y-adapter(the side of the y-adapter with dual female connections). Run another single male-to-male rca cable from the extra audio out on your tube preamp to the other female rca connection on the y-adapter(the side of the y-adapter with dual female connections). Lastly. run a singl male-to-male rca cable from the single female rca side of the y-adapter to the rca input on your subwoofer.
This set-up will feed the sub/LFE signal to your sub when using your processor for ht and it will send the signal from your new preamp to your sub when listening to 2-channel music through the preamp. You may need to adjust the sub's high pass filter to find a 'happy medium' cutoff frequency that works well for both ht and music. Or, you could change the sub cutoff frequency depending upon which you're listening to. I usually set my sub to about 35hz, which works well for music and ht for me. Since your left and right front channel speakers are very low frequency capable, you may also want to set your laft and right mains to 'large' on your processor settings to direct the full frequency range to your front speakers. 'Large' is the setting on my Parasound processor that sends a full range signal but your Audio Refinement Pre2 DSP may be called something else.
To summarize, I think a tube preamp with 'ht passthru', with an output impedance of 4,900 ohms or less and with an extra audio output for your sub should be a very good fit for your system. I know that the tube preamp I use (VTL 2.5tl) does meet all these criteria. However, I think there are probably several others that would work just as well and would sound as good or possibly better. I'd suggest seeing if a local dealer would let you try a couple out in your system.
I'm not exactly sure why you want to buy a new processor since the one you have is a very nice unit. The only reasons I can think of are either you don't like the sound of yours or you want to use one of the newer codecs.
But, if you have extra funds burning a hole in your pocket, I would suggest you consider replacing your Adcom amps with some newer class D ones (Wyred4 Sound, D-sonic, ClassD Audio, Channel Island Audio, etc.) This would give you a bigger bang for your buck, sound quality wise, as long as you're satisfied with your current surround sound capabilities.
If you're set on a new processor, You may want to check out the new Emotiva processor that has gotten very good reviews.