Unfortunately all passengers on the same booking must be travelling on the same flight and class.

As your FT status shows BA Silver, note you can guest only one person into the lounge. So if both you and your partner are both Silvers (or Gold), then it doesn't matter that the two older children are in a separate booking, each of you can still guest one in.

But if your partner doesn't have BA Silver (or Gold), then the ticketing strategy should be:
- ticket first child in Club
- ticket your partner in Club
- ticket yourself in WT
- ticket second child in WT
That way, first child and your partner can access the lounge based on their Club ticket.
And you can also access the lounge based on your Status with second child as guest.
Once you get onboard, you can swap seats with first child, so you can sit with your partner in Club.