if by "seller keeps AR & APP" that he/she is responsible for collections while you are dealing with future sales/deliveries (hence the customer deposits being yours) I can see this getting very messy and undermining the future of the business. The AP issue could also be very risky if the seller is required to make the payments and then doesn't. It is essential that customer and supplier contacts only flow through the company regardless of the ultimate beneficiary/accountability.

In terms of AR i would (a) have the collections managed by the company; (b) i would have a clear list of invoices going to the Seller's accounts and in fact deposit them there. At the same time I would escrow/reserve part of the consideration to cover AP putting that in an account to make payments. The amount can be set to the amount due within the next 60 days. AR collections can be used to top up the account if it drops below a certain level.

happy to speak if that would be helpful

1. If you are undertaking the responsibility for the work associated with the customer deposit(s), the seller needs to leave you the cash in the bank (hard money) to cancel out the obligation they are giving you.
2. You need to control all customer relationships. While the seller may kept the economic benefit of all AR recorded as of closing - you need to be the one to control the collections. AR Oldco and Newco money should go to you or be under your scrutiny, and then you decide the legitimate split. Should be no quarrel here - it's all fair and legitimate. What if the customer does not specify what the payment is for? Is it old money or is it new money?
3. Regarding #2 above, cut-off and AR recognition policies need to be scrutinized. Make sure the seller is only taking the money for the products they sold or the things they have done.
4. All Oldco AP needs to be paid off at closing by the seller. If they refuse, you need a list with specific dates of intended payment - all to your satisfaction and subject to your confirmation. How the seller treats the vendors who remain post-closing is very important. When things get rough, it will be this family of vendors who are or who are not your friend. Many states also have bulk sales laws that buyers need to be aware of (i.e. either seller notifies its vendors of sale prior to closing or buyer can be held responsible for seller's unmet obligations).
5. What about utility deposits (other), credit card processing, ACH / wire processes - both customers (incoming) and suppliers (outgoing), etc. It is not as simple as they keep the old bank account and now you have a new one.
6. Each one of your suppliers will set your new entity up with a new customer a/c#. What does this entail? Do you have to share info. or make applications for credit? This process takes time; need to make sure your stocks on hand are covered by functioning supply channels.
General - you have a lower middle market acquisition (read: attractive) structured like the sale of a what is referred to as a "main street" business (e.g. laundromat). Your transactions terms should include normal working capital measurement and settlement. With the mechanism you have, only you can loose. The Seller knows all of the things you do not know. What's the EV / terms?

I cringe when I see a post like this. Samer and Walter comments are right.
a) You are very likely overpaying if you let seller keep A/R, A/P and assume customer deposits (I presume seller is keeping cash). If value has been adjusted for this, make sure you can fund WC post-closing. [A customer could ask you to fix a problem of past before giving you new business. And, a vendor can ask you to pay off previous invoices before shipping new raw material.]
b) You should control customer relationship, even if seller is keeping A/R. You collect and pay the seller.
c) If seller keeps A/P and does not pay, vendors have legal rights to the inventory that you think you paid for.
d) You should get a matching asset for the customer deposit liability.
e) Consider assuming A/R & A/P, and adjusting price if needed. Then protect yourself through R&W in APA for quality/quantity of A/R, A/P.
f) Asset purchase or Stock purchase is irrelevant.
g) Customers will get confused if they get calls for seller for old A/R and from you on new A/R. Vendors will hold up your shipment if seller does not pay them. Best solution is you control the relationships.
h) By letting seller keep A/P, you are giving up interest-free vendor financing.

It is usually best to agree on a working capital target based on a normalized level of positive working capital. At close, if working capital is below the target you reduce the purchase price accordingly (or take it out of an escrow account). If working capital is above the target, then pay the seller for the additional amount using your LOC.

Agree with all of the above.

Further, one of the assets you're hopefully purchasing is the right to keep using the name!

I agree

Most efficient way is not to consider this sort of asset/liabilities split, way too risky ! Why would you let your clients and suppliers temporary in the hands of the seller? Evaluate the company as a whole, get only rid of the excess cash and buy the shares (after extensive due diligence done by professionals and a sellers' warranty).