I would assume that only 187.5k (75% of 250k) will be available for you to borrow, so you'll only likely get 87.5k from your HELOC.

What are people's thoughts on borrowing from your 401k to get the initial capital?

A HELOC is what we call an "Open End Mortgage" that is money can be drawn down up to the loan limit by the borrower.  

I am always glad to answer any financing questions.You can borrow for up to 10 conventional mortgages! 

PROPERTIES LISTED FOR SALE For a rate and term refinance transaction, the borrower must evidence that the listing has been cancelled, and must not have been listed for sale as of the date of the application.

The borrower must have sufficient funds to cover borrower-paid closing costs and fees at the time of settlement.

Yes, the Bank will see it, primarily because as a borrower you are required by federal law to disclose the other loan obligations on your application for a mortgage loan with them and if you don't it' a federal offense and a felony in all 50 states.  

Debt/Income Ratio - this could be a problem as you indicated... good thing about FHA is that you can use family money as a "gift" for a downpayment so I would recommend the same thing Cory did above with borrowing from your parents and paying them back using cash flow.3.

Where are the remainder funds coming from that you aren't borrowing?

This is Freddie's way of rewarding borrowers that worked with the lender on a short sale rather than just walking and letting it foreclose.