Sacrificial Construct says:
[Trash]: Prevent an installed program or an installed piece of hardware from being trashed.
Yes and no.
While you are free to use Sacrificial Construct to prevent sacrificing your Clone Chip or your Shiv, this means the cost was not paid and nothing happens, other than the Sacrificial Construct being trashed.
Compare Clone Chip with Acacia: Acacia will still work and can be saved by Sacrificial Construct, Clone Chip will not, since Clone Chip's cost was prevented.
If the cost of a paid ability requires a card to be trashed, then preventing that card from being trashed prevents the cost from being paid and the paid ability does not resolve.