mgmt: mcumgr: Fix Bluetooth transport issues

This fixes issues with the Bluetooth SMP transport whereby deadlocks
could arise from connection references being held in long-lasting
mcumgr command processing functions.

Note: Heavily modified from original PR due to differences in MCUmgr
operation since the Zephyr 2.7 release.

Fixes #51846

Signed-off-by: Jamie McCrae <jamie.mccrae@nordicsemi.no>
(cherry picked from commit 2aca1242)