Lassen Sie uns... would be used in a formal context, like a Professor addressing his students using
Lassen Sie uns diesen Sonderfall mal genauer untersuchen...
Note sich etwas tun lassen is already reflexive - If you combine this with another reflexive verb in infinitive, things start to sound awkward:
Lassen Sie uns uns unterhalten!
Lassen Sie uns uns vorstellen!
Even if grammatically entirely correct, most people would avoid this awkward construct and evade into something like
Lassen Sie uns reden!
Machen wir uns doch kurz bekannt!
Stellen wir uns doch untereinander kurz vor!
(The doch here is a particle that comes quite close to the "Let's" meaning)
As a side note: Translating "Let's" as lassen Sie uns is not always optimal - there is a wide choice of possibilities that often tends to fit better
- Wollen wir ... ?
- Können wir ... ?
- Sollen wir ... ?
- Mal ... (sehen. [for let's see])
- Los, ...
- a simple infinitive ("Fahren wir" for let's drive)