cross database trigger: server principal is not able to access the database under the current security context
This should be fairly simple to accomplish via module signing. Look at my other answers here on DBA.SE:


Permissions not allowing Update or Select on a trigger executed function
Permissions in triggers when using cross database certificates


The basic concept is as follows:


Create a Certificate in the Source DB (i.e. where the Trigger exists).
Sign the Trigger using ADD SIGNATURE.
Create that same Certificate in the Target DB.
Create a User in the Target DB from that Certificate.
Grant that Certificate-based User INSERT, UPDATE permission on the table in the Target DB.