Your battery's serial number was never the same as your MBP's serial number, and there is no way for you to tell whether another battery was substituted for yours in the shop unless you have a record of the original battery's number. Your MBP's original purchase receipt and packaging would not have included the serial numbers of components and subassemblies like the battery — only the whole computer's serial number.
Batteries sometimes swell abruptly as they approach the ends of their useful lives, and it's entirely possible that your battery did so while it was in the shop for a month. Now you will need a new battery as well as a new logic board if you want to continue using the computer. But because you aren't willing to pay the cost of a logic board replacement, the battery doesn't matter.
If an iPad would meet your needs, buy it.

I like this idea overall, but Apple really ought to include some sort of check digit ('https://en.wikipedia.org/wiki/Check_digit') mechanism, and publish the formula they picked, so that it's possible to determine with high confidence whether a supplied Apple serial number could actually really be a genuine Apple serial number. (Only Apple would know for sure, of course.) There are all sorts of real world scenarios when businesses, governments, and other organizations need to take accurate inventories and when somebody transposes a digit or otherwise gets the serial number wrong. It'd be darn helpful if all these organizations could know, right away, when they've collected a serial number in their inventory taking that cannot possibly be a genuine Apple serial number.
Also, Apple ought to eliminate all characters from serial numbers that can be easily mistaken for other characters. This generally includes (as examples):
0, O, D, and (probably) Q
5 and S
1, 7, and I
8 and B
U and V
The choice of typeface can matter a lot here -- for example, to distinguish A from R -- but some easy-to-confuse characters ought to be dropped no matter what the typeface. This'll help even Apple's own support teams over the phone, etc., as they deal with consumers. There are still plenty of characters available to allow plenty of serial numbers for billions of products.
Finally (and now that Jony Ive has left!), don't be afraid to make serial numbers easier to read (in larger type) on devices. Apple has got typographic experts, so spend some effort making serial numbers easier to read within whatever (sometimes lame) aesthetic goals Apple is trying to achieve. IBM ThinkPads were pretty damn stylish, and they had big, easy to read serial numbers in easy to inventory places.