Can you get on to any litigation files as part of the firm's family law practice? It sounds to me like building up a base of litigation experience - in whatever field - would be an asset to getting the kinds of jobs you want.
Also, if the 'business law' area includes any employment law, there may be a potential path to human rights law through there.
I echo what Deadpool said. There are a lot of lawyers in policy roles, such as in the PM and EC pay groups, trying to break into the LP group. Unfortunately, internal postings for LP-01 positions (generally, lawyers with 0-4 years experience post-call) are very rare because the DOJ fills LP-01 positions with their own articling students and the SCC/FCA/FC/TCC law clerks through unadvertised processes. As a result, there’s not much use for internal only competitions and if they need additional people they’ll generally run an external competition.
Everyone I’ve heard of who transferred from a government policy role into the DOJ did so through networking and getting an unadvertised appointment. So if you take the policy job and want to get into the DOJ, do not rely on getting in through a formal internal posting.