Two questions and two easy answers:
Question #1: No (but see Note 1 below)
Question #2: No (but see Note 2 below)
Note 1: Although attempts have been made to make this work, they all have employed extra joints or goofy sliding sleeves over elongated tone holes. So, there's no workable simple solution by modifying the clarinet. However, there is the time-honored "string down the bore" trick.
Get a relatively thick piece of twine about as long as your clarinet's bore. "Fray" out one end of the twine, then drop the non-frayed end down the bore from the top of the barrel. Install the mouthpiece while trapping the frayed end of the twine in the socket, allowing the twine to fall down the bore.
I have actually tried this (with a variety of thicknesses of twine until I found the right one). It works (sort of), dropping the pitch of the horn down to the A clarinet levels, but it plays like - well, like a clarinet with a string down the bore.
Note 2: The terms "cheap", "inexpensive" "functional" and "A soprano clarinet" are never found together. To begin with, and excepting the cheap Chinese instruments, makers have only produced them for their intermediate line of horns or better. So, we're talking a significant investment for most. Second, the used instruments that find their way to eBay are going to go for a premium, since there are many others also looking to score an "inexpensive A soprano clarinet".
You could go the Chinese route. There is at least one A clarinet from the Forbidden Kingdom that is on the market today. However, I've looked at a number of Chinese clarinets from the top tier, and I wouldn't touch them with a ten foot pole.
Now, for the inexpensive solution to your "play along problem". Someone (I don't remember who, but others will) makes an inexpensive C soprano clarinet. The keywork is somewhat simplified (and all made of plastic - the keys actually snap into place on the joints), but intonation and workmanship is first rate. It's made for smaller children taking up the clarinet, but it would suit your situation perfectly.
Don't look for these second hand - they're a specialty item in the extreme. But, for playing straight off of a piano or vocal chart, they can't be beat.