Consider the client, the market, the benefit they will receive from the use of the tune, what competitors they have that may be interested in it if they do not make an agreement that is satisfactory to your needs.
Remember, up front money is one thing but long term income is much more desirable.
And remember, NEVER, NEVER sell the rights to ownership of your work, and never give exclusive rights to your work unless you are compensated properly, also provide a sunset clause in your agreement such as 3 years which requires them to renegotiate with you for continued use of the work for an additional X years.
If the tune is a great one, is used widely and creates a following then after they use it you may find other outlets for your work. So if you give exclusive life long ownership to others, you lose out.
I'd be glad to look over their offer or yours at no charge or cost if you'd like. You can keep the proprietary info out of it as far as I am concerned.
Congratulations! You are on your way or you have already been there.