However, there are private lenders that may offer low down payment construction loans and consider projected rental income.

So you can have $5M in equity but if you have zero income then they will not give you a loan.A private lender may provide financing but its owner occupied so that gets thrown out.If you are over 55, you could get a reverse mortgage - that is really your only other option. 

They are my preferred lender if I'm not using private capital for a deal.

I also know some private lenders that lend on land with lower LTV's - of course more expensive than a bank but it can be done. 

I’m looking to find all sorts of people to work with, contractors, agents, private lenders, staging businesses.

Yes, hard money lenders will absolutely lend on a luxury home renovation of this size, but there are some key factors to consider.Loan Size & LeverageMany hard money lenders have loan caps, but larger private lenders and funds specializing in luxury properties can go up to $5M+ per deal.Typical Loan-to-Value (LTV) is 80%-85% of the purchase price.Some lenders also offer After-Repair Value (ARV) loans at up to 70%-75% of ARV, which could allow you to leverage more capital.Loan StructureIf you’re using hard money for acquisition and self-funding the renovation, lenders will likely structure a bridge loan with interest-only payments.If you wanted renovation financing, you’d need a fix-and-flip loan, which has a draw schedule based on project milestones.Terms to ExpectInterest Rates: Typically 8%-12% depending on experience and deal risk.Points: Expect 1.5%-3% at closing.Loan Term: Usually 6-18 months (extension options available).Down Payment: Depending on the lender, you may need 15%-25% down.Luxury Market ConsiderationsExit Strategy: Lenders will be cautious about absorption rates in high-end markets.

But the other options are Hard Money Lender, second on a house, personal line of credit, private money from an investor, or PM from a family/friend.

Alternatively, you should start contacting private money lenders and/or hard money lenders so that when you find the deal, you are ready to go with acquiring the money. 

Also, I see that setting it up as a private mortgage is an option as well?

As an avid public and private markets investor, I solely believe a person's network is their net worth.