"tmsh save /sys config file" command to save the full config of a Link Controller ?

Link controller is a cut down version of BigIP DNS (previously called BigIP GTM). The config requires a few GTM objects that in version 10.x were stored in wideip.conf. In version 11.x, gtm config is stored in bigip_gtm.conf

So whatever applies for GTM also applies to LC.

If I create a few GTM objects - a datacenter, gtm server, gtm pool, and wideup, then create an SCF (save sys config file gtmtest), then inspect the resulting file /var/local/scf/gtmtest, I can see all the GTM commands are present in there.

You could also use tmsh to list the config objects that you're interested in, and then use "load sys config merge from-terminal" on another BigIP to paste that config into it, without having to bother with editing the SCF