Headphone amps with 'link outs' are just passthroughs of the input line out signal.
I have the same need as you on one of my computer rigs and use a
Crane Song Avocet going out to my powered monitors... but I'm presuming anything like this is not realistic for your proposed budget.
Best bet if you're looking for a one-box solution at a low price is a used preamp with a headphone socket. Not that many around, but doable. Or cheaper still buy an audio switch and connect the headphone amp / powered monitors to the outputs and your computer to the input. If new, Taiwan-based
Firestone Audio apparently makes the 250 quid-ish 'Little Country' which is not on their site, which is a tube preamp/headphone amp. Shoot them an email and you should be able to find out whether it's available.