If you are moving from your SFR to another property, then you should be able to get a 5% down loan or even 3% if it's FHA.  25% is only if you won't be living in the property in my experience.

I they accepted an offer of $118K from my brother but since he couldn't get the loan, we're starting over.

Home 1 is ideal for a fix-and-flip loan to fund the remaining $90K, which I can help you evaluate options.

The only way to pay down a loan, is to pay down a loan.

While you work on that, you could also explore creative financing options like DSCR loans or partnerships to fast-track your first STR.

The two existing LTR units (on my family’s property) were built 5–8 years ago at a cost of about $80,000 per unit, so I’m estimating a 20% increase in costs for the new builds.My plan is to use the equity in the land as the down payment for the construction loan, which I expect will make closing costs the primary upfront expense.

yes, you could use the HELOC as a down payment on a hard money loan, and then the hard money loan for the rest.  

You can get a normal owner occupied home loan to do this. 

Many investors use new construction loans or bridge financing before refinancing into a long-term DSCR loan.

At first blush it looks like you're finance at $500k, so getting a loan for $375k.