The BS4 basestations are currently connected via UP0/E interfaces to the Unify PBX. Unify can handle the devices, manage endpoints, etc.
(Up0-interface - Wikipedia)
Is there some kind of hardware available to keep the basestations and integrate them into a new Asterisk setup?
I believe that this is not easy to achieve, if at all possible. As I understand it, the roaming and handover process is controlled by the Unify system. That is probably quite proprietary.
A better solution would be to replace the base stations with DECT multicell systems that are directly SIP-compatible (e.g., Snom M900, Gigaset N870 IP PRO). It may be possible to use the handsets with the N870 (officially, only devices from the IP Pro family are supported). However, the base station “speaks” DECT GAP. This is not the case with the M900.