If Im understanding correctly, youre buying the home from your mom who will take a first position lien on the property in exchange for carrying the $100k of the purchase price on a private note. You're also trying to raise another $100k to rehab the property to get it leased up.
If your mom is wanting to keep the note in place once the property is stabilized for personal income, I would probably cashout refi your other rental that has equity and use that cash for the rehab of the new property. This is likely the cheapest and simplest pathway. If your mother does not want to carry the note long term, there are GSE renovation loans that can be used for both purchase, rehab, and permanent financing all in one loan. Another option is hard money for the purchase and rehab, which is paid off with a Rate/Term or cashout refi afterward.
I would not have her deed the house to you so that you can borrow against it if she's selling it to you and not giving it to you. This will create a huge mess. I also would not use your heloc unless you plan to cashout refi one of the properties once the rehab is complete - if you dont cashout refi one of them, you wont have any way of extracting the rehab funds to pay off the heloc.
The only other options I can see are A) your mom partnering with you rather then selling the property to you - you become 50% owner and bring the funds and management to the rental while she provides the existing equity as her contribution; or B) your mom lending you the rehab cash out of her personal funds while also seller financing the house to you.
I doubt that any serious lender is going to lend you construction funds in second position behind your mom's purchase-money lien on a distressed property. If anyone does this, theyre likely doing it as a favor and not a for-profit venture.