Gorgergrat and Matterhorn Glacier Paradise in the same day?

Time wise both are definitely doable.
There is the weather concern factor though.

It is fairly common for mornings to start out clear and then clouds and fast moving thunderstorms move in the afternoon.
If that happens you would not want to go to Klein Matterhorn in the afternoon ; of course the weather in this mountain region is highly unpredictable and totally out of your control so best not to worry about it and give yourself 2 full days so you can adjust on the fly as needed. Some days have the opposite weather pattern than above.
From the Gornergrat the primary thing to see is the Matterhorn, could be a day with only a little bit of clouds but if in the wrong position it can obstruct the Matterhorn view which kind of ruins it.

I would probably go in reverse your plans as I believe the lifts to Gornergrat run later than Klein Matterhorn.

If you have time on the way down from Gornergrat ; get off at Rotenboden and walk the trail / Riffelsee hike to the next stop ; Riffelalp. Very scenic and an easy trail mostly all downhill. That to me is the highlight of the Gorngergrat area.

You really don't need a hat, gloves, long underwear, etc...
A normal pair of hiking pants is good, along with a tee shirt, fleece pullover and lightweight jacket for August. ideally something waterproof jacket wise. Wool socks are a good plan, they are good in the cold and also the warmer weather, thin ones are fine for August. Always think in layers, so you can lose the jacket or lose the fleece layer easily as needed.
Only parts actually cold are inside the Glacier Paradise at Klein Matterhorn and you will not be there for long. If you do a couple of runs on the snow tubing you will be fine dressed as mentioned if you want to ski for hours up there you may need more weather wear.

Since we brought our 5 year old daughter we did bring gloves, hat and snow pant layers for her but not us, she enjoyed playing around in the snow, throwing snowballs and riding the snow tube many times at Klein Matterhorn.

Don't forget the sunscreen!

Thanks so much for responding. Just the information I needed!

If you do both in 1 day ; buy a Peak Pass
Head up to Klein Matterhorn in the morning, then on the way down get off at Furi ; not Zermatt and take the Riffelberg Express (is a small cable car) from Furi to Riffelalp ;
then the train from Riffelalp to the top of Gornergrat
From top of Gornergrat you can take the train all the way down to Zermatt or do the walk I mentioned as part of the way down.

I would guesstimate this would save 1 hour in transit time alone (2 reasons, the 2 lines are not that close in Zermatt, requiring you to walk through the whole town to get from the cable car to the train but right next to each other in Furi and Riffelalp ; so saves on walking and the other reason the Riffelberg Express takes a couple of minutes to reach Riffelalp vs. about 25 minutes on the train to reach Riffelalp)
With a 1 day peak pass everything I mentioned will be included.

You can get lunch in Riffelalp.

Thanks so much for the information - very helpful!!