Ah well, it was only for a day, the little strips of holy paper I was asked to swallow along with some water saw to that . Miss Cometa seems to live in the same Universe as your cat! (I LOVE the link, thank you!).

Even if you don't try and get to the front of the crowd at the very end, it can still take a long time to make it through to the shrine before you can get out (you get coralled through a strict one way system) But I don't know what the point of doing this would be if you don't throw some coins and pay your respects at the end you might as well just visit Meiji Jingu on another unrelated day. I've never found the crowds at the end to be that hard to navigate through (they keep a channel open through the middle and there's no problem at all if you stick to the sides).

In periods where Tokyo ladies backcombed and lacquered their hair, it was usual for them to come home and shake out lots of coins. Just wear a hat and don't look behind you - it's not that bad.