Connection lost for no reason at all.

What Logitech support might have missed is a broken Bolt usb receiver. Since you have another M650L (let's name it mB) with another bolt receiver (rB), use Logi Options+ app to pair the non working mouse (let's name it mA) with second receiver (rB). Do it other way round as well (mB-->rA). If you can rule out that both receivers are not broken, then probably its a broken mouse. If both mice are working and one of the receivers are broken, you can buy a replacement bolt receiver for around 10 dollars and have two working mice. You can tell this feedback to Logitech and they sometimes can be generous and send you a replacement receiver. If your broken mouse is under warranty, they would do RMA and replacement would be sent. You have to provide them proof of purchase for this. Without this document, they would deny your claim. Sometimes, Logitech could be notorious for not having replacement receivers, then check amazon, ebay and AliExpress for replacement receiver. MAKE SURE TO USE NEW ALKALINE BATTERIES FOR TESTING. I knew various instances with even good battery the mouse refuses to pair. It would pair with new battery only. Reason is, it has to broadcast the pairing wireless and takes more than normal amount of energy for tiny micro controller of the mouse. Let me know the findings.