> In this case what is a good SDR device that does TX?

The hackRF is a popular one. If you want to do anything at range, note that you'll need an RF amplifier and an antenna.

Also note that TX is heavily regulated, and you should get licenced before you start blasting.

> Is there a very low bandwith equivalent of Internet via SDR?

There's open source software to build your own LTE tower using an SDR. That's not really low bandwidth, though...

Most of the SDRs are receive-only. The ones that do transmit have really low power. The LimeSDR is 10mW and most are in that range. That means need amplifier or two to make it usable.

Packet radio is done with amateur radio. There is both VHF/UHF ones that use normal radios, and 2.4GHz ones which usually use Wifi with dishes. There aren't BBSes because there isn't much point with Internet, and there isn't much point to Internet access when have mobile. Plus, it isn't allowed to provide non-amateur Internet access.

There's no BBSes now, no. There once were, I often used F6FBB's amazing BBS software though I never ran one full time. But many people did. We had all night chat sessions at a time where phone BBSes cost serious money to use.

These days packet is boring as hell, nobody chats anymore. It's just aprs position messages that nobody bothers to look at :(