One thing you could do is to write a small check from the fund and offer the rest via SPV to your LPs as a perk. You can use Decile SPV for this because it’s built for this use case and is managed in hub.

Important: Capital lines of credit from your bank are NOT an option in this scenario because you have not closed the fund. Banks are not going to provide your a line of credit due to the risk and it's not cost effective.