Reminds me of my Sony Walkman that liked to beep for every new song. There are BT units available without these sound effects.The info so far doesn't include the most important part that is giving you the problem. Apple had three versions of the Airport Express with some mayor differences with the last version. Is it MC414LL/A ? As to Bluetooth your iPad Air 2 and Pro both have BT V4.2 but your Logitech BT is V3. It's OK but 4.2 or 5 would be better. The iPad's unfortunately have a limited Apple codecs choice but for break music should suffice. The thing to avoid is double compressing such as using MP3 files to play via Bluetooth on a iPad using AAC. A lossless source file send over BT would be best such as ALAC or FLAC. I am using a pair of BlitzWolf transmitter/receivers with aptX HD codec but my need was for distance and quality. This is overkill for break music. I would have suggested using a iPad that was not needed for loading iPadOS13 and not for your upcoming gigs. I see that you took the plunge, good luck.
The Airport Express I am currently is A1392, though I tried my my older one as well, which I cannot find right now to give you the version, but it was from when I first got a Mackie DL1608, whenever that came out.
I have the latest firmware on them that they will take.
They worked fine up until a month back when the new IOS came out. Previously they were both slow to connect to AirPlay from any iPad, taking 5-6 seconds, and sometimes reverting back to the iPad's sound.
But after IOS 12x, it was taking 10-15 seconds sometimes to connect and also taking one to 4 tries to get it to stay connected.
Then randomly disconnecting during playback, making them totally unreliable to play music for cutting the cake etc.
I'm already having to keep up with 3-4 devices to charge, I don't really want to add another one to cover what I have been doing fine for years.
So the bluetooth adapter is working well, a little laggy, but not bad - at least it never disconnected, and the bonus if it connecting as soon as the power is applied without me having to do so as I always had to do with the AirPlay is NICE... LOVELY in fact.
I tried researching a better Bluetooth adapter that would give me the latest standard, and NOT make connection noise, but I cannot find one...
iPadOS 13 seems MUCH faster btw in switching between my music player and the Mackie MF. So far, so good.... If something should go wrong, we do have old iPads on standby 8-)
I'll let you all know what happens after this weekend (two gigs, a rehearsal, recording a video all in the next 4 days)
Appreciate the input btw!