Let's answer your question on its own terms
No, you can't solve the problem by making the sphere larger. All that does is reduce the amount of heat absorbed per-square-meter (density of heat absorbtion on the inner surface). It slows the process down, but doesn't solve the problem because you need to absorb 100% of the star's energy over its lifetime anyway.1
Others have mentioned using heatsinks to get the heat from the inner surface to the outer surface. That seems so 2018. I believe that 100% of the sun's emissions are photonic,2 let's put that to use.
Layered solar panels such that each layer captures a different wavelength spectrum, substantially increasing the efficiency of energy capture. This would reduce a ton of the heat because you don't need, what, 99.99999% of the surface of the sphere? But you do need to put the energy to use or, well, boom... you know... boom.3 Think "stick a capacitor in a light socket" boom, but bigger.
Massive pinwheels in space that are driven by the solar wind. This converts mass amounts of solar energy to kinetic energy. It has the benefit of creating night/day zones on the inner surface. You could build giant poles to hold them in place, or just use rockets to do so (If you have the energy to spend building a Dyson sphere, you have plenty to spend doing this).
You could let the heat build up and then use massive thermoelectric generators to convert excess heat to electricity. Your world will be a bit more like the Dagobah system, but hey... Swamp Thing needs a place to hang, too. Like the first idea, you need to do something with the energy afterward.
Finally, just because I'm feeling whimsical, you could make your sphere out of coupled cylinders that gently spin the sun-facing-side to the outside. This might be hard to visualize, but think thread spools laced together into a sphere, each spool spinning. It would play merry havoc with your gravity, but then... zero-G during the day and 1G at night? That might be pretty cool. But it solves the heat problem by allowing the excess to irradiate into space like any planet does. (Now that I think about it, this would allow for a common atmosphere inside and out with the ability to wave to your friends on Cylinder XZ21N-7 as they rotate past. You'd have storms build up between cylinders.... dang, that could be fun...)
BTW: The fact that you have no gravity inside a Dyson sphere is becoming well enough known that you should explain to your audience how you're solving that problem. It's an off-topic comment as the presence of your population is irrelevant to the question, but since you mentioned it, please keep that in mind. Besides, you need to do something with all that excess energy. You might as well use it to drive your artificial gravity generators.
1 If you make the radius so large that the issue of heat absorbtion becomes irrelevant, the value of building the sphere is lost both in terms of its construction cost and the inability to use the star's energy effectively.
2 If I'm wrong about that, please let me know. Thanks!
3 That link is an esoteric joke.... But I like the tune.