You could try applications of sulfur. It is most effective when tilled into the soil but seeing how that is not an option for you, I would apply it after core aeration. Core aeration will also help with your heavy clay.

When would be the best time to core aeration? In the Spring or Fall?

Can you buy sulfur, if so where and how much to put on?

It will be very difficult to lower your soil pH over a broad area like a lawn using sulfur. I used either a 25 lb or 50 lb bag (sorry, but it has been a few years and I forget how large the bag was) just to do the area around the drip line of a tree. I also used a bulb auger to drill holes a foot or two deep and poured the sulfur into the holes. Spreading sulfur on the surface doesn't work very well because the process that converts sulfur to acid does so by first changing it to sulfur dioxide (a gas), then combining with water. If it's on the surface, too much of the sulfur dioxide is lost to the atmosphere. It might work if you do it just before a rain and it dissolves and goes down into the soil.

Another thing to consider is why the soil's pH is that high. If you're on soil that has a lot of limestone in it, even if you lower the pH with the sulfur, the effects will probably be short lived. Is high pH soil common in your area? Usually, soils east of the Mississippi tend to be acidic. If it used to be acidic and has been limed too much, the pH may go back down on its own over time.

You might want to try to increase the organic matter. That might bring the pH closer to neutral and also help by buffering the pH. Mulch mow when you mow. When you're not adding seed, you can mulch mow your leaves. I wouldn't want to mow now that you have seeds down, though. Depending on the size of your lawn, you could try fertilizing with used coffee grounds from Starbucks. Just fling them around the lawn; they don't need to be spread exactly evenly. But they're a fairly mild fertilizer, so if your lawn is large, it's not very practical. Some people use ground grains as fertilizer.

Since it can be so difficult to lower the pH over a large area like a lawn, you might want to try a few things that will help the lawn even if the pH stays high. You can spray the grass with a water soluble iron. With water soluble iron, it is absorbed directly into the leaves, so the high pH doesn't interfere with its uptake. Another thing that can help is greensand (AKA glauconite). It's not available everywhere. You can have it shipped, but that would be very expensive. I've never used it (I can't find it around here), but I've heard that it can be very effective. It's slow release, so it takes a while to get started, but the effects last a long time.

I've found sulfur at a local nursery. If high pH soil is not common in your area, it might be difficult to find sulfur. If anybody has it, you should look for soil sulfur. If I were you, I think I'd try some of the other approaches first. I also probably wouldn't do anything drastic unless I saw signs of chlorosis (yellow leaves, especially yellow leaves with green veins, although I'm not sure the green veins applies to grass).

Not sure about out-gassing of sulfur. Applied to soil, sulfur is converted to sulfuric acid by soil microbes in the presence of oxygen and water. You do not want to apply too much at one time because of potential damage to the lawn. I have applied about 4 lb/1000sq.ft (5lb/1000sq.ft of the product linked below) in consecutive springs and did not notice any negative effect. It may also take a while for the effect to be seen. I attached a link of the product I used--it is 80% S with some iron and manganese added--probably a good product for you. I would also agree about BPGreen's comments made about organics and foliar iron. Over time, it is possible to lower pH--the big question is will it last or will you have to manage the pH as an ongoing issue. I am sure your grass can do well at that pH but it may not be all that it could be. Given that you spent some $$$ buying the best seed, sounds like that might be important.

Here is a link that might be useful: Disper-Sul

What was the condition of the grass before you overseeded? If you had good healthy grass before I would not be too concerned about the Ph, it may adjust on its own with good basic lawn care.

What has been your fertilizing routine? Chemical fertilizers will tend to reduce the Ph over time so that could also be in your favor. Look for a slow release X-0-0 (Lesco 39-0-0 as an example). You have plenty of phosphorous and potassium.

"Chemical fertilizers will tend to reduce the Ph over time so that could also be in your favor."

I've read that before, but I think it's highly overrated. It's possible that it can be an issue for acidic soils, but it doesn't seem to help alkaline soils very much. My experience has been that increasing organic content moves an alkaline soil toward neutral much faster than chemical fertilizers.

I think it depends on which chems--some directly lower ph when dissolved, some have sulfur coating for long-lasting effect, some have no effect on ph at all.