How to get smooth motion when changing clip speed?
Shooting at a higher shutter speed is not an option in my opinion, as it just blurs the motion until the problem is not visible.

I don't get that part. Isn't hiding the problem until it is no longer visible what you want?
Also, a higher shutter speed results in less blurred frames, not the other way around.

I quickly figured out what is happening: by increasing the speed, my 30 FPS video becomes 30*1.2=36FPS. But I am exporting at the standard 30FPS so every 6th frame is dropped.

This is the default behaviour of the Media Encoder. Adobe calls it Frame Sampling. As you said, since frames will be dropped at regular intervals, it will produce curious results.

So finally my question is what option do I have to get smooth motion or hide this problem when changing the the speed of the video by small intervals?

Use one of the other Time Interpolation modes Premiere Pro has to offer. At the bottom of the Export Settings dialog, there's a dropdown menu containing three options: Frame Sampling, Frame Blending and Optical Flow. You will find the same dropdown menu in the Speed/Duration panel that can be accessed via the context-menu to change the interpolation mode of individual clips.
From the documentation:

The Optical Flow feature in Premiere Pro uses frame analyses and pixel motion estimation to create brand new video frames, resulting in significantly smoother speed changes, time-remapping, and frame-rate conversion. [...]
Frame Sampling repeats or removes frames as needed to reach the desired speed. Frame Blending repeats frames, and it also blends between them as needed to help smooth out the motion.

Here's another helpful article on using the Optical Flow interpolation.
I suggest you try all the interpolation modes and see what works best for your footage.
One more thing, you probably don't want to hear that, but 30 fps video is really bad for doing slow or fast motion effects (except maybe timelapses with 200% speed or more). As you explained yourself, there's not that many frames to go on so some frames will need to be dropped, synthesised or blended in some way which looks subpar at best. The interpolation algorithms do a decent job, but it will never look as good as it would with footage that was shot at a higher frame rate in the first place.