Certainly possible.
But.
Time will be against you with the plan you have.
If you can get an early train (set an alarm or three) from Luzern (I don't think you said what time of year this would be) over the genuinely beautiful Brünig pass you can be as far as Interlaken Ost by - say - 8 or 9 o'clock on the 06:06 or 07:06. Then straight up the mountain to Jungfraujoch.
If you wait later, or if you hike first and train up the mountain later, you will likely run into fog or clouds. What can be brilliant sunshine in the morning often starts to cloud over at Jungfraujoch in late morning and often by noon is completely socked in. If you go late you'll miss the good stuff. If you go late you also get the masses of tour bus crowds up there and on the Jungfraubahn.
I disagree that car is as good as train over the Brünig. It may be for some of the passengers but the driver will have their eyes on the road and the traffic unless you stop at some of the scenic lookouts, but there goes any speed advantage. Also the road mostly one lane each way with no overtaking (except motorcycles of course to which no rules apply). I have before now been behind a slow tour bus the whole way.
The train is a cog railway and has virtually seamless beauty the whole way (sit on the right leaving Luzern and stay in the same seats when it changes direction in Meiringen (the home of the real Reichenbach Falls which you can go up) for the best scenery). It is true that about half way down the pass there is a view where you are high over the valley and looking down for the lake and the waterfalls, for your height averse, but the train is extremely stable and smooth and I don't know anybody who has had trouble.
The car, on the other hand, has many single lane each way tunnels on the route (can't see much inside tunnels except the magnificence of Swiss tunnel manufacture) and as you drop down into the plain between Meiringen and Brienz you have a series of very steep and very tight corkscrew turns and switchbacks, The slope up to the pass from the Luzern end is gentle and not always noticed (except by the lookout and rest area before the summit) but steep and technical going down (reverse of course on the way back).
I know the previous poster is local, and I have only crossed the pass maybe 15 times, split between car and train, and, yes, for much of the route it covers similar territory, but I maintain that the train is much more dramatic, much more impressive, and much more fun.
So - early train; up the mountain early (check the screens before you ascend); hike when you come down with whatever time you have left. The scared of heights among you might skip any cable cars like the one from Grindelwald to part way up the Jungfrau, and think how they will handle the road if driving or car passengering..
