First of all, it seems that you are solely concerned about the Funding Valuation Adjustment (FVA) here, and not CVA; Sovereigns have credit risk which should also be valued here given they would not be posting any collateral as mitigant when the market moves in your favour. But let's focus on FVA:
It is important to think about FVA (and all other VAs also) as the adjustment to a transaction's PV as the result of moving from one state/set of settings to another. Because dealers will typically hedge themselves in the inter-dealer market, the swaps they enter into to offset new client trades follow standard market conventions, assuming perfect cash CSA hence OIS discounting.
Specifically in your example, assuming you would hedge yourself in the open-market like a dealer would do, your FVA comes from the fact that, when the swap PV goes into your favour, you do not receive this value as collateral, but on your hedge, you need to post it anyway. Hence you would need to raise funds to be able to fulfil the collateral call on your hedge.
However, if your hedge is uncollateralised (unlikely to happen irl) then the FVA would result from you needing to post to the Sovereign when the MTM is negative for you but not receiving it on your hedge. I think this is the scenario you had in mind given your point 2.b)
Assuming this latter scenario, the standard way to value the FVA cost is close to what you describe in 2.b), but not exactly; simulate the paths and calculate the cost of funding incurred for raising funds on the expected negative paths (i.e. Expected Negative Exposure, or ENE). You would calculate it based on your internal treasury's funding curve and probably take out this funding day 1 as a static hedge (as a strip of forward-starting borrowings). This is what you would need to add on top of the price of your uncollateralised hedge.
To answer 1., I don't think it is a good approximation, but it would depend on the nature of the swap, its maturity etc., and most importantly on what you would qualify as good.
On 3. I see what you are saying but you need to be careful not to get mixed up between the legs of the swap (pay vs receive) and the sign of the MTM (value against or in your favour). The synthetic replication you mention does not seem correct as it is the net of the legs' cashflows which is discounted in a swap, and a one-way CSA does not mean that collateral will be posted on one leg's PV only, but on one side (positive or negative) of the net PV of both legs.
If you have a correlation built-in between your simulated market and funding cost, apply the latter to all the simulated negative paths and then average it, otherwise you can apply your funding spreads to the average of the negative paths.