If you want to be generous and offer some kind of settlement, that's up to you. 

They offer a nice discount off the settlement fee.

Banks can't get out of their own way in order to close faster than 30 days.This is quite true, but what is somewhat ironic and comical is that the bank as seller will find an offer with a short closing time to be a stronger offer than the offer with a longer closing time; then once under agreement the settlement will get extended as needed by bank asking for amendment to the agreement via an addendum.

Lien holder offers settlement in lieu of me fighting their redemption.  

If not - we will go into second stage negotiations to get a better price.Once we agree that the price is right - that is low enough to make an equity profit - we will than use my pro-active auction marketing system (one that I have used for over 20 years) to sell the property  - all cash in as is condition with NO contingencies with settlement in 30 days.We will advertise the property for sale for 3 weeks and than have a public auction - so as an example if we control the property for $100,000 and sell it for $120,000 - that means we make $20,000.How much do you make if we are successful in finding an assignee? 

(Note to self: Never close on the end of the month if you can avoid it) So that lease, and the deposit, wasn't included on the HUD-1 settlement.

Also, I am confused on how the financing and settlement process works.

I got mine free but they may have a smaller fee.From that you can make an offer, if there are costs to obtain clear and insurable title deduct those from your offer and describe the liens that need to be cleared by you at settlement.

Title companies are in the business of issuing insurance, providing settlement services & acting as an escrowee in certain circumstances. 

That just isn't going to happen.If a tenant willfully wants to leave and wants to settle the obligation for the remaining term of the primary lease for a discounted amount then generally a landlord has prepared by an attorney an agreement to vacate,lease termination, and a settlement agreement.No legal advice given.