According to Wikipedia, ッチ is rendered as tch, not cch (etchi not ecchi). Logically I would think that ッジ would then be dj but there's no example of it. What is the correct way? dj or jj? Is there any comprehensive guide to Hepburn romanization that helps with this?
cch => tch is the only exception defined in the Hepburn romanization system and its variants (such as パスポート式). All the other 促音 (
っ) are expressed by repeating the same consonant. Specifically, I'm not aware of any romanization standard which uses
dj. So if you want to strictly obey the rule, バッジ would be rendered as bajji even in the Hepburn system which is thought to be closer to English pronunciation. In 訓令式 this would be bazzi. IMEs accept both.
Traditional Japanese words occasionally have っち (eg, 八丁堀【はっちょうぼり】, 坊っちゃん, 湿地【しっち】), but probably there's no word that contains っじ/っぢ. Perhaps that's why there isn't any standard. There are many katakana combinations which are used to write loanwords but are not strictly defined in the traditional romanization systems (eg, ツァ, ティ, ドゥ). They are not suitable for writing non-Japanese-origin words in the first place.