I bought tiny (13.6mm x 16mm, sold without header) bluetooth-to-uart module from LCSC (0.84 euro, stock 7000+,
https://www.lcsc.com/product-detail/C20539408.html) and I originally had some troubles making it work, I searched here too and found nothing so once I made it work I decided to put it here so that other can find it.

The SOIC-8 chip is JieLi AC6328A (32bit mcu with BT and USB support), chip marking is BP2P064-28A2 (classic bullshit JL numbering that means nothing). The oscillator is 4pin 24MHz. I searched for "I6328A schematic" and found something very similar, pinount matches: 1=vcc, 3=gnd, 4=antenna, 5,6=osc, 2 is used for led. It is programmed, I found chip itself on aliexpress, they sell it programed and unprogrammed as well, but it's not much cheaper (0.7 eur) so I wouldn't bother using ali.

When I inspected pcb before first use I found small blob of solder under the part connecting the antenna, I measured it and there was short, I desoldered it and the short was still there (approx 2 ohms), RF is weird. I soldered it back on, still 2 ohms. I installed android app "nRF Connect" by Nordic semi, it found the device as XLBLE and when I connected it and sent string via service 0xFFE0 and characteristic 0xFFE1 I was able to read it on uart at 115200. Then I implemented bluetooth in my own android app and it also found it and sending strings also works.
I'm powering it from 3v3 that supplies CH340N and connect GND pin a TXD from the module goes to RX pin of CH340N. For app I just use serial monitor from arduino ide at 115200. Green led is on when device is connected. I didn't test receiving, don't need it.
It consumes 5mA when idle and 10-17mA when sending data. Range is at least 5m (width of my apartment, can go further than that).