That's because Classic theme doesn't use PNGs, it's internally rendered, right? Anyways, this is... beyond ridiculous if you ask me. Tap tempo button is really useful.
:) Anyway, it's possible to create a fixed classic reapertheme. I attached the needed image folder to the post. It just contains one image (which is a single transparent pixel ;)) and a config.txt with a single line (to switch png usage on). Unpack this folder (called classic_bpm_fix) into your ColorThemes folder. If you don't know where it is, go Options -> Show Reaper resource path in explorer/finder. Open Reaper and load the Classic theme (if not already loaded). Make sure "Default image resources" is set to "Classic UI" and "Allow themes to override" is ticked in Preferences -> Appearance -> Theme Editor. Now check the box labeled "Theme uses image resources" and choose "classic_bpm_fix" from the dropdown next to it. Hit apply and enjoy the regained peace in your transport bar:) Don't forget to save your new theme with a nice name.