In short, no. You can tell the customer whatever you want on your webpage, at your point of sale, or in the memo field of a BIP70 invoice---but once the transaction has been sent to the network, there's no way for the customer to take it back. (Except for double spending, but that's discouraged.)
In the long-term, miners will likely change their transaction-selection process to allow for child-pays-for-parent fee calculation. For example, imagine merchant-created unconfirmed transaction B spends an output from customer-created unconfirmed transaction A. Currently, most miners use a naïve formula for selecting whether to include those transactions in their block attempts:
- If transaction A pays a high enough fee per kilobyte, include it in the block.
- If transaction A is included and transaction B pays a high enough fee per kilobyte, include transaction B as well. (Note, consensus rules don't allow including child transaction B before its parent transaction A.)
In child-pays-for-parent, the miner will do the calculations above the same but also do a third calculation:
- If transaction A's fee plus transaction B's fee is high enough per kilobyte of the combined size of transaction A plus transaction B, include both transactions in the block.
This allows merchants to effectively add a fee to any low-value transactions, helping to encourage miners to confirm those transactions quickly. This method does have some overhead, so having the customer pay an appropriate fee would be ideal, but it can help prevent transactions from getting stuck.