There are two kinds of credit risk: jump to default (JTD) and the CDS spread delta (CS01).
If you're long a corporate bond, and you bought CDS protection on the sovereign, and the corporate bond defaults, then you don't have an effective JTD hedge. So let's just focus on CS01 hedge.
Assume for simplicity that all the bonds are USD-denominated and that you have these components:
You can observe the sovereign CDS spread (but perhaps not the corporate CDS spreads) and the bond quotes.
If you can't observe the CDS spreads for some of the corporates, you can proxy, for example as sovereign CDS spread + corporate - soveregn z-spreads.
A CDS model can compute CS01 by tenor bucket.
A bond model can compute the bond-CDS basis (given bond quote and CDS spread) and also CS01 by tenor bucket and interest rate deltas by tenor bucket.
A P&L explain (P&L attribution analysis - PAA) attributes the P&L from CDS to the changes in CDS spreads, carry, rolldown, interest rates (IR - very little for CDS); and the P&L from CDS in terms of all these and also the bond-CDS basis. (You can further reduce unexplained P&L by including CDS spread gamma and various cross-gammas.)
The daily PAA will show how much P&L came from CDS spread movement (you can choose the CDS, and occasionally adjust so little net P&L would be here), interest rate (which you can also hedge if you like), bond-CDS basis (which you cannot hedge with CDS), and the additional credit spread of corporates over sovereign (which you cannot hedge with sovereign CDS, but could hedge with corporate CDS if they trade).
To show the effectiveness, you can backtest (like historical VaR) - show what the P&L would have been over the last few years of historical market data and/or generate many Monte Carlo scenarios (like VaR) for what can happen to your market data; and see what the P&L would have be umder all these market moves, and how much of this P&L is attributable to what.