Yes it is ok to send. I will do this whenever an item is overdue and it is unable to be located straight away. Few days overdue I will express post, a week refund the postage amount and after that partial refund as well. 

Buyers are always grateful but many will say not to refund as they need the item.

Contact the buyer, and put it to them.

If you give them a choice, partial refund of $??,  or  cancel transaction.

If the buyer then goes with option B,  you are entitled to  cancel the transaction, with the reason at buyers request, and therefore no blame or consequences  are attributed to you.

Edit

Just to add, if they choose the partial refund,  you will be required to ship, and  your sales metrics will receive a defect for late shipment.

By the way, how late in shipping are you.

It doesn't sound like a great idea to me. If I were the buyer, I'd be wary to accept that offer.

The reason I'd be skeptical is because you didn't send it once and I'd be worried about accepting only a partial refund when I had no guarantee you would actually ever send the actual item.

If you feel bad about the transaction, then apologise and refund in full and tell them you'll still send the item for free. Then follow through.

Or else just apologise and refund in full & keep the item.

Making a partial refund then sending an item that is only a few dollars is a bit risky for you too as the buyer could claim item not received unless there is tracking, and I am guessing that at that price point, there may not be.