The dramas with a bromance theme I have seen is school 2013, Shut up flower boy band, Sungkyunkwan scandal, I Am sorry I Love you, I remember you, Rooftop prince, Flower boy next door, She was pretty, Kill me Heal me, Sassy go go, reply 1997.
Sassy Go Go has a really strong bromance in it, and so cute! Also has romance too though, so I don't know if that's what you'd be interested in. Others are Bromance School 2013 I Remember You Shut Up Flower Boy Band (one of my absolute faves!) Reply 1997
Off the top of my head (all are J-dramas unless otherwise noted):
Battery *Beach Boys *Ikebukuro West Gate Park (though it gets dark at times and since OP doesn't like violence/fighting, probably not for them) *pretty much any Kamen Rider series (some *cough*W*cough* border on straight-up romance rather than "bromance") *Kisarazu Cat's Eye *Mahoro Ekimae Bangaichi Nobuta wo Produce(though I see the OP has already seen this one) *Ping Pong (J-movie) Samurai Sensei *Stand Up!! *Tiger & Dragon Tumbling *Water Boys Youkai Ningen Bem
(though I like all these-- I wouldn't recommend them otherwise-- the * means it's an especially-loved title)
A Gentleman's Dignity - it's a must watch for bromance lover :) beside the romance between the male lead and the leading lady it's all about friendship of 4 men, watching those four in action made me cry from laughing...
The Bridal Mask - very complicated relationship between two friends and those two had more chemistry than the "main couple"
Bad Guys - it's about the most dangerous criminals who are after other criminals, the relationship between those three "bad guys" is very interesting and it develops well during the show
