Well, I don't know anything worth of mention about welding for a chassis, sorry for not being able to contribute. I make holes when welding, so...
Anyway, I know a bit about structures. I think your assumption about stiffness is incorrect.
Yes, after a quick check, an elliptical section has around
one third of the stiffness
in compression of a round section, for very similar cross section areas and, hence, weights.
The failure, in the exercise I'm looking at right now, at Bresler & Lin & Scalzi's structures text, occurs by buckling. I'm talking about columns and the such, where loads are more or less symmetrical.
If you're using a member that works in pure bending, you could go with the elliptical section, aligning the long axis of the ellipse with the loads, altough if that's the case it beats me (please, understand I'm civil) why you wouldn't use an I-beam (or a box, for torsion). But if it works in compression, in principle, forget it: you're chosing a worse section.
Trying to think as a mechanical guy, I'd bet that elliptical tubes could be used if you're thinking of buckling them in a crash: even in my simple book that's mentioned (for joints). Anyway, I also think you should be aware of the stiffness you're giving away in compression, and incorporate it into your design.
Don't do anything based on my limited knowledge of structures, please. I'm used to security factors most guys here would laugh about.