Is it permissible to complete a marriage without a guardian if the family refuses to marry off the younger daughter before the elder due to an "unjust custom," and what is the legitimate solution for this issue, and to whom does guardianship transfer in a country where there is no Sharia judge?

A summary of what pertains to you, your role in the problem, and the solution you are part of:

You should try to convince the family to change their custom and allow you to marry their youngest daughter. You can seek help from close advisors.

If your attempts are unsuccessful after a reasonable period, then give up on this girl and look for another who is compatible with your circumstances.

Leave the matter of withholding and its problems to the girl and her family. If the girl feels the problem, she will seek a solution herself.

And "And whoever fears Allah - He will make for him a way out And will provide for him from where he does not expect. And whoever relies upon Allah - then He is sufficient for him. Indeed, Allah will accomplish His purpose. Allah has already set for everything a [decreed] extent." (Al-Talaq: 2-3).