I wanted to spend less than $100, too. Several times. I've got plenty of remotes that almost work. I ended up with a Harmony 880. It would have been cheaper in the long run to have just bought it first, but of course, it wasn't for sale when I first started looking.
The 880 is best for HT use. And my JVC D-ila set normally "cycles thru" inputs, but does not have to. There are codes that will direct it to the specific input you want in one step. I suspect your set is the same. Harmony has all that on line, pretty much invisible to you if you wish. It just works. You tell the system what gear you have, what is plugged in where, and it will turn on the right equipment, to the right settings.
Make very sure you look over any potential remote, for a button for each purpose. Most especially, make sure there is a button to be remapped for the skip ahead and skip back functions. On my 880, I use the >>| and |<< buttons.