Term for lending an item to a store so they can sell it on your behalf, for a fee

Solution 1:

If a retail outlet accepts goods to be sold, on commission, and to be returned if unsold, then it is commonly called a 'sale or return' arrangement, although this may be applicable only in the UK.

Sale or return link is to the British 'Financial Times':

"Definition of sale or return - If a shop buys something on a sale or return basis, it can return the goods it is unable to sell"

Formally, it is referred to as consignment.

The detailed methodology is discussed on business forums.

The ownership of these goods remains with the sender. The agent sells the goods on behalf of the sender according to instructions. The sender of goods is known as the "consignor" and the agent entrusted with the custody and care of the goods is known as the "consignee".

Wikipedia :

Consignment is the act of consigning, the act of giving over to another person or agent's charge, custody or care any material or goods but retaining legal ownership until the material or goods are sold. That may be done for the purpose of shipping the goods, transferring the goods to auction or intending the goods to be placed on sale in a store (a consignment store).

[Edit after Post regarding comment : Anyone following the link on my profile will see that I am a Director of my own Publishing Company - Belmont Publications - and I am familiar with retail practice in the UK.]