Is mulch or topsoil better for new lawn in clay soil?
Assuming the new topsoil is decent topsoil, the topsoil is "better". Even if you had outstanding topsoil now, additional topsoil would still be better because it would be a thicker layer of good topsoil. You simply can't have too much topsoil.
I'm trying to imagine what you have now and I'm picturing a muddy mess with tracks from heavy equipment and probably much of the topsoil is gone. Since it's described as "clay", I'm also picturing a high-rainfall area (40-50 inches annually). If my guesses are true, then the dirt is acidic, lacking in calcium, low in organics and will take a long time to bring up to par. Lime takes at minimum 6 months and most likely 1-1.5 years to add calcium in useable form to the soil.
I'm having to make a lot of assumptions to make a recommendation, but a soil test is a prudent place to begin, as OrganicLawnDIY said. In addition to CEC (or TEC which should be at absolute bare minimum 10-15), pay particular attention to calcium % and hydrogen % in the base saturation table. If calcium is low and hydrogen is high, you may as well get new topsoil. Even with the new topsoil, I'd still apply lime to the soil you have now, then cover with the new soil. Lime is fairly cheap. If you look on craiglist, topsoil isn't too expensive... about $30 a pickup truck load in my area.
I also agree that tilling is going to be a lot of work with little gain and will result in humps and bump and lots of weeds. It will break up the soil for a while, but it will go right back to its compacted nature since the problem of compaction is due to lack of calcium, which tilling won't fix and neither will mulch. Mulch (or compost) will add friability to the soil until the mulch decays, then you're right back where you started in terms of compaction.