Your fund size should be the minimum target. So in this case $10mm is the minimum target and you can oversubscribe it to a hard cap of say $15 or $20m. The first thing that you should do is to see the conversation through with the LP and get them to a close. There is a long way to go between a verbal yes (10% chance of closing) and a wire.
Until then, it's not worth it for you to solve this issue. For example, it's possible that the LPs may want to invest under special unfavorable conditions that you should not entertain. For example, an LP may require that you invest their money in a certain geography or type of company. In this case you would need to refuse this LP because the ask it's not viable operationally and has major ethics issues.
Now let's say they do decide to sign and commit. In this case, let them in for a smaller ticket so that they don't own more then 25% of the fund. Then, as you oversubscribe the fund, they can increase their position.
Until then, it's not worth it for you to solve this issue. For example, it's possible that the LPs may want to invest under special unfavorable conditions that you should not entertain. For example, an LP may require that you invest their money in a certain geography or type of company. In this case you would need to refuse this LP because the ask it's not viable operationally and has major ethics issues.
Now let's say they do decide to sign and commit. In this case, let them in for a smaller ticket so that they don't own more then 25% of the fund. Then, as you oversubscribe the fund, they can increase their position.