You could use these funds to reimburse at least some of your cash. 

(Or if we transfer the money after the fact to reimburse the partner who paid upfront.)Our lender is accepting of a deed transfer to a land trust.Thanks!

Had some issues with the HVAC and plumbing which they reimbursed me for the repairs/charges.

Your partnership agreements should clarify that your partnership is not responsible for reimbursement of such expense  and he could treat it as such. 

Two ways to do this: 1) You have in the partnership/operating agreement that partnership reimburses for the expenses incurred by the partner individually. 2)You deduct your Unreimbursed expenses at your individual level as an Unreimbursed partnership agreement (UPE).

I reimburse myself from the LLC each month for the addtional mortage costs. 

Would it also be a good idea for that 1 sibling to pay for the legal probate fees and somehow get reimbursed on the back end?

Say something like you can reimburse them 200 bucks for their time.

Even if the lender is covering the renovation costs, they usually work off of a draw schedule where you perform the work, and then you are reimbursed.

Yes if you have a good CPA and reimburse money from your LLC to your personal.