I have been using the Woodpecker coping sled for years with consistent results. It is easy to use and pretty accurate. There is an updated version that is well outside your target price range, but the original one is still available, and right on budget.

https://www.woodpeck.com/woodpeckers-coping-sled-3029.html

Here are a couple links from Woodcraft: 

Deluxe Coping Sled

Coping Sled

I bought one of the Deluxe ones but haven't taken it out of the box yet.

Peter

I had settled on that same Rockler sled (it got good reviews and I have free shipping).  But I am going to go back to the table saw method. 

With the router I had a hard time getting grooves and stub tenons perfectly centered.  I then had to sand the surfaces flat. 

With the table saw, the tenons and grooves are guaranteed to be centered.

I use a heavy cast iron tenoning jig that makes relatively quick work of the task. 

The two outside blades of my dado set are just under 1/4” so two passes easily makes the 1/4” groove.

The tenoning jig holds its setup (I don’t use if for anything else), so that is quick and easy.

And the faces are always flush—no sanding require.

If I decide to get a coping sled, that Rockler one is what I will be ordering.

I always did it that way too.  I used a 12” x 12” piece of plywood.  But lately I have not been getting clean results and I don’t know why.  In the past, it always worked perfectly.

As an aside, if you ever retire a wooden interior hand rail, cut off a couple of 6” long pieces and screw it to the push material (easier with my 12” x 12” plywood).  It offers excellent hand purchases and is very comfortable.  You can buy it by the foot form Home Depot, but I had an old used rail.  Set it on a slight angle towards the fence to guarantee good contact with the fence along with the forward movement.

Just outside of your preferred price range, but I have not found a coping device better than the Aigner Contermax. I originally bought mine for an older router table, and still use it regularly on my shaper if I don't want to bother setting up the outrigger.

https://martin-usa.com/products/contermax/

An interesting design.  I wrote to the company for pricing and purchasing information.

Amusingly, the black handle that they have on this sled is almost certainly identical to the black handle I purchased (but never used) for my barn door.  Instead I made it from 1” diameter oak dowel and 1” wide, 10 ounce leather belting. 

https://www.ebay.com/itm/185528506337?chn=ps&_trkparms=ispr%3D1&amdata=enc%3A1fBwUswEhQAy8vWRizgXepg60&norover=1&mkevt=1&mkrid=711-117182-37290-0&mkcid=2&mkscid=101&itemid=185528506337&targetid=1583904486842&device=c&mktype=&googleloc=9004882&poi=&campaignid=15275224983&mkgroupid=131097072938&rlsatarget=pla-1583904486842&abcId=9300697&merchantid=459454543&gclid=EAIaIQobChMI5s_2qIfZ_QIVia3ICh2PVQLYEAQYAyABEgKlNPD_BwE