Audio Research makes (made) a good integrated with remote control which you should be able to find used for your price without a problem. It's a damn good match with ProAc Response series speakers which, if you're real lucky, you might find a pair of 2.5s for $2,000.00. I once heard a Manley Stingray integrated sound nice with Coincident floor standing speakers.
But to answer your question (even though most everyone else already did), yes - you can find a tube integrated which will play your speakers as loud as you want. I've been doing same for 20 years. Actually had a Mesa Baron which in pentode (although a bit sterile sounding) will blast music and can "rock a party". However, with the right speakers they can do this in triode for a much smoother, more natural sound. The Baron can be listened to in full pentode, 2/3pent-1/3tri, 1/3pent-2/3tri or full triode. They are actually two mono amps on one chassis but you'll need a pre-amp (or even a used Creek remote volume control for $150.00 if damn nice).
But to answer your question (even though most everyone else already did), yes - you can find a tube integrated which will play your speakers as loud as you want. I've been doing same for 20 years. Actually had a Mesa Baron which in pentode (although a bit sterile sounding) will blast music and can "rock a party". However, with the right speakers they can do this in triode for a much smoother, more natural sound. The Baron can be listened to in full pentode, 2/3pent-1/3tri, 1/3pent-2/3tri or full triode. They are actually two mono amps on one chassis but you'll need a pre-amp (or even a used Creek remote volume control for $150.00 if damn nice).