I'm trying to decide whether to use the market value of its assets or the book value
when lending money to a company.

The book value is gotten from a pre-determined method of depreciation that follows the policies of the company. There are different methods and therefore the book value could be different things for different companies, even if it's the same asset with the same market value. How a company happens to choose to depreciate something says nothing about their actual financial standing nor their ability to pay a loan.

So what you are saying is the when trying to decide whether to use the market value of its assets or the book value
when lending money to a company, that I can use either?

As lending is backed by fixed assets as collaterals, I safely assume the asset to be falling under IAS 16's jurisdiction.

As per IAS 16, one can use either a cost model or a revaluation model for asset recognition. Under the revaluation model any discrepancy between the book value and the market value is promptly addressed, thus rendering the choice upon the management to revalue the assets or otherwise.

Having seen the above treatment the lender can freely assess the loan-worthiness of the enterprise considering post and pre revaluation capital employed with the proportion of interest bearing debt.

When a potential lender has to make a decision, he is interested in:
1. Whether the borrower has sufficient assets for collateral security?
2. Whether the borrower is in a position to pay the interest and installments as and when due.

In order to satisfy the first condition, the borrower's assets would be assessed at fair value by an independent valuer and for the second part, the lender will evaluate the performance of the past five or more years supported by the projected financial statements for the next five years.

There are other non financial matters that play a role in decision making.

The question is simple!
The answer simpler still!

Preferred method:-
MARKET VALUE. (Meaningful)