I have no problem with the spelling shier, and it's probably the one I would use.
The general rule with adjectives ending in y is to form the comparatives and superlatives by changing the y to i and adding -er or -est: happy, happier, happiest.
The reason why shier is a bit weird is twofold: 1) the y in shy is pronounced [aɪ] instead of like in happy, so you could argue that it doesn't have to follow the y -> i rule described above (as if it were a different phenomenon), and 2) shier looks a lot like pier, which is pronounced as one syllable like peer, [pɪr], instead of like pyre, [ˈpaɪəʳ].*
However, I personally see no problem with treating shy just like happy, despite the different pronunciations; and moreover, regarding point (2), we do indeed say & spell supply -> supplier instead of supplyer, and no one ever mispronounces supplier as *suh-PLEER.
*pronunciations based on AE