You asked about doing a 1031 of “your portion”. 

If you wanted to fund a portion of the down payment would you suggest a hard money lender, personal loan, etc.?

Once you have completed the consolidation portion of your 1031 exchange you now own a more expensive (and probably really nice) property that it must be your intent to hold for investment use. 

I’ve done some research and have found that some banks will allow the seller finance portion from the seller to go towards the down payment of the loan, is that true?

Quote from @Andrew Katz: My tenant paid a portion of their rent for December on 12/31/24 (late).

Since I'm taking such a cash hit, I asked the sellers agent if they would be able to designate some portion of the sale price ($200k?)

Lenders often include a portion of the cash flow in the investment property as income to you personally to offset the DTI. 

I need funds (but only for the buyout portion of the deal, and she's agreed to certain percentages).

So if we were to assume the loan, we need to take a second mortgage to cover the equity portion of the loan.

Additionally, your brother would inherit your cost basis for the gifted portion, potentially leading to higher capital gains taxes if he sells later.