If I understand what you're asking and the beam is old and only A36 then it can support 780 pounds per linear foot, and since only half of the 8' span is on it that's 195 PSF with about 1/2" of deflection in the middle.
Send it. It would even work up here.
I should mention that load is for tolerable deflection according to aisc 9th edition, and still a ways from yielding. The beam is also more likely to be 50ksi unless it's ancient.
You should match or exceed your columns and their supports to the strength of the W-beam if you can.