Using a sub with no sub out on my amp

You can also just run leads from the speaker to the sub. It won’t effect your sound unless you are a bat with super hearing. 
Not all subs take this type of connection. Rel is a brand that does. 

Id save more money,250 isnt going to get you anything worth buying .Get a used Rel t5i and run it high level output off your speaker outs

Thanks guys for the replies I'll look into that suggestion see what I can find one for.  Either that or I just look for another amp.

Your amp is fine. 2 options- buy a sub with speaker level inputs, or add a Line Out Converter to run a sub at line level off your speaker terminals. Either way no changes to the amp and won't affect the amp.

A LOC is just a couple 10k to 20k resistors that drops speaker output down to line level. They are under $20 at Parts Express, or you can even DIY for only about $2. That's what I did, see the pic on my system page. https://systems.audiogon.com/systems/8367?utm_campaign=website&utm_source=sendgrid&utm_mediu... If its good enough for me....  
With subs don't bother spending a lot on one it is better to use more, but one is better than none- 2 is better than 1, etc.

Don't worry about which channel either. Seriously, I have done it all these different ways, adding subs to an amp with no sub out is the easiest thing you can do.

Luckily, it’s not the amp, it is the speakers that are lacking in bass. Any self-powered sub with speaker wires in/speaker wires out will solve your problem.

I just added this 800 wpc self-powered sub to my small office system.

https://www.rockvilleaudio.com/rock-shaker-12-wood/

On sale, in your budget, 90 day trial with free return shipping (I bet you keep it like I did).

It gives a nice fullness to my bookshelf speakers. Don’t over-do it, just a bit, un-noticeable until you turn it off.

Your speakers are 89 db sensitivity (better than my Wharfedale Diamond 225 87 db)
Your amp is 30 wpc, more than my Luxman’s 10 wpc.

Your woofers are 5-1/8" diameter, they can be great but can only move so much air. Mine are 6-1/2", same thing.

When your sub’s amp takes the low bass job away, the bookshelf speakers then do a better job with upper bass/mids/highs and thus imaging improves along with detail ... no distortion from trying to do what they cannot do.

You will be happier at lower volumes after you add a sub to those speakers I bet.