Hate to sound harsh, but, in my understanding, no-tripod is a common knowledge among Japanese photographers who shoot well-known Buddhist temples in Kyoto or anywhere else in Japan. In Kyoto, aiming your camera at Buddhist statues (and surrounds) inside the main halls is prohibited even if that's done in a hand-held posture. Tripods are out of use anywhere inside their precincts, since their use is not allowed either, in the first place. Not very sure, but they don't refer to the use of monopods. But I can't bring myself to suggest trying your luck. Ten to one, staff will come around to say you can't do that. A gorillapod needs something to hold onto, you know. So you'd probably need a handrail/pillar/tree branch.., again staff will come holler "Don't do that!" But I hope that doesn't let you down. In other situations, you are Ok photographing in a hand-held posture.
Likewise, in Kyoto, most if not all well-known Shinto shrines don't want you to take pics of stuffs inside the main halls, (or disallow it), let alone bringing in tripods to their precincts. Yet, then again, taking pics of other things is perfectly Ok as long as images you get are all hand-held (and you keep good manners with you).
Btw, am I right to assume you're not a commercial/professional photographer? (If so, then you are refused to aacompany your clients into the said temples/shrines. So you don't see pros there schlepping their tools of trade: reflectors, sun shades, light bulbs, and etc, including their assistants. In case you see some, they are affiliated to/under a contract with the temple/shrine.) If not, don't overthink, just follow the rules. You sure capture nice images, I gather. Yes, that takes patience to do. Golden Pavilion, Ryoanji, terrace of Kiyomizu, Fushimi Inari, Arashiyama river gorge and Tenryuji plus the bamboo lane, Heian Shrine garden, Nanzenji, to name just a few, will all be in your photo album.
Lastly, should you take your tripod with you, let me suggest you squeeze in a visit to Nara (a 30-something mins train ride away). The deer park and surrounds are an A-OK place for your photo shoot. No one complains. Yes, photo is Ok of the great buddha statue, but no tripod, no flash.
Have a blast!