Is the husband considered negligent or sinful before Allah for allowing his wife to spend lavishly despite financial hardship, and for accepting her work to help him with life's burdens?

As long as you spend on your wife and children in a reasonable manner, there is no sin upon you. You are not obligated to spend on them beyond the obligatory maintenance, even if you are wealthy. It is even more appropriate, then, that this is not obligatory upon you when you are unable to increase your earnings. There is no harm in your permitting your wife to go out to earn money, as long as her work is permissible and her going out does not violate Islamic law.