Send the text to your phone, then forward the text from your phone to your customers, so they will see it's from your phone.  Forwarding texts is just a few extra taps.  Some phones may also let you "share" the message with other messaging apps like WeChat or WhatsApp, if your customers are on those platforms.  In other words, rely mostly on your phone's capabilities for this.  Sending directly from PC to cellphone networks is the most convenient, but only if it works.  The tech industries have made it hard to do so, and will continue to make it hard.

I'm pretty sure Twilio has a service where you can get a custom number that's just yours when you send using their service. But it has honestly been a while since I've looked into their service offering so you might want to talk to them about it. But last I knew, you could do that; you got your own SMS number that you can send in and receive texts from and it would look like it was coming from you.

Mark What adjustments in the code did you make to able to send SMS using Twilio?