Private car or small group day trip tour from Lucerne to the Lauterbrunnen Valley area

You can go on an organised tour. Google will show you those.
You could look for a private hire. But that will be expensive. This is a country where hiring professionals is expensive. This is one reasons why such service are not that readily available.

But in both cases the experience would be less than going by train. I would really suggest you do that.
- The train ride itself from Luzern to Interlaken is scenic.
- By train you can make interesting combinations. For example do part of it by lake steamer. Do a circuit Interlaken - Lauterbrunnen - Wengen - Männlichen - Grindelwald - Interlaken
- You could fit in a hike etc...

All things that are easier if you do not have a car bogging you down..

I guess part of it may come down to if you want to see beautiful scenery on the way.

The train between Luzern and Interlaken Ost is one of the most scenic rides in Switzerland. It is a cog wheel narrow gauge highly scenic route, waterfalls, lakes, mountains, wildlife. If scenery is important the train is the answer.

The highway which is roughly parallel now spends most of the time that it is near the scenic towns and views in tunnels. These tunnels have been built in order to bypass the intermediate towns to speed up the journey for trucks and other vehicles at the expense of the scenery. If all you want is speed (not hugely better if you get behind a slow bus or truck, it is almost all single lane each way with a solid line preventing overtaking, or a traffic jam which I have more than once) then maybe an expensive driver would suit.

Size and ages of the "small group"?

It will only be faster if your driver uses the route through the tunnels. It looks like there are at least 2 long ones—-the Sachseln Tunnel and the Giswiler Tunnel—between Luzern and the Brunig Pass. I don’t know how you feel about tunnels, but I hate them.

For considerably less than the money you would spend on a car and driver (which I am guessing would run into hundreds of francs), you could get a scenic and relaxing train ride there and back, and a lot of fun riding the lifts between Wengen and Grindelwald.

https://www.maennlichen.ch/en/

You could have a nice lunch at the restaurant up there before riding the gondola down to Grindelwald to explore the village.

https://www.maennlichen.ch/en/summer/information/fare-prices-from-grindelwald.html

Well I don't know if you have any experience of driving in Switzerland.... but here is how we doing things here:
- Many large towns and cities regulate the number of cars that can enter their area and as a consequences you could find yourself sitting at traffic lights waiting for cars to leave before you are allowed to enter
- Roads are maintained by the individual communities not the states so it is possible you may be delayed by several different maintenance crews on the roads
- Many communities active discourage the use of cars by limiting the amount of available parking space, limiting the parking times, regulating the traffic flow or an outright ban on cars

I don't know if you'll be faster by car, but I do know you'll have better scenery and more reliable travel times if you use public transport.