We know Jeremy broke and entered back into the house for 'papers'.
By most people's standards, having been released on bail, this was a foolish act.
Police believe it was his last attempt to cover up some tracks, since he knew he was now a suspect.
Tracks such as leverage marks at the window (which if he had broken in AFTER the murders, he could plausibly explain).
Regarding the silencer, there's a lot of ways to go on that score...
IF he did it, he could have taken the silencer with him - but, if he was caught with it, it would look bad. If it was noticed missing - that would look bad. If ballistics showed a silencer was used, and it never turned up - it looks REALLY bad and suicide definitely didn't happen.
If the silencer is found at the house, then great this works for Jeremy, but not when it's in the cupboard.
IF the silencer was next to Sheila's body, that looks pretty good... Sheila uses the silencer, then removes it to kill herself (making it easier to reach under her chin).
If the silencer is found in the cupboard, this looks bad for Bamber, as it seems less likely Sheila would have put it away as part of the suicide.
One possible explanation is that he didn't want it to be discovered missing - so makes sense to leave in the house.
He didn't want to even look like it had been used - so makes sense to put it back in the cupboard (but would have to be sure it was clean OR hope nobody would examine it, since the scene looked like a suicide anyway, and there was lots of gun stuff around).
The down side to this theory though is that he had to place ultimate faith in forensics NOT being able to know a silencer was ever used. As soon as they know that, they'd look for it.
In the end, forensics didn't know a silencer was used, and when they considered the possibility that it had been, they still couldn't prove any bullet went through it.
So did Jeremy do a bad job of cleaning it? Could he have known that backspatter might put blood inside the relatively narrow end of the silencer?
Or was he just a 25 year old who thought he could beat the system - and lost.