Maybe I can help. Essentially, there are six categories of backings. I will describe each for you, and then make a few comments on your proposed plan:
AIR
This is the safest and easiest backing there is, provided your board/stave's back grain is sufficient to safely undergo tension.
BACKINGS THAT ADD PERFORMANCE
These backings are designed to do one or more of the following:
A) add draw weight,
B) increase a bow's safe draw length margin by allowing short bows to be relatively overdrawn,
C) allow for more dramatic profiles that would be unsafe on a selfbow (these profiles are generally induced at glue up),
D) allows the bowyer to use each wood/material to it fullest potential based on its characteristic(s) (for example, hickory is fantastic under tension and makes a wonderful backing, but is relatively weak in compression and so makes a relatively poor belly.)
E) allows the bowyer to create a "recipe" that can readily duplicate performance, draw weight at a particular length, etc (this is much more difficult when building a selfbow.)
F) allows the use of wood (specifically in board form) that has a violated grain on the back that would otherwise fail.
COMMON EXAMPLES: bamboo, hickory, fiberglass,
BACKINGS THAT ADD SAFETY/PROTECTION
These backing are designed to do one of more of the following. They generally don't add performance or draw weight:
A) prevent a splinter from raising on the back of the bow by creating, in effect, a continuous layer that simulates a single growth ring
B) protect a vulnerable material from the elements (an example would be covering the back of a sinew-backed bow with snakeskin or bark to help keep precipitation off the moisture-sensitive sinew.)
COMMON EXAMPLES: silk, linen, paper, rawhide,
BACKINGS THAT RAISE THE NEUTRAL PLANE
These backing raise the neutral plane of the bow (i.e. that part of the bow that neither goes into tension or compression.) What is does is allow more wood to go into compression while the backing takes over the majority of the tension work from the back. The most common example is a cable bow, which is simply a bow that has a taught cable of sinew, silk, linen, modern bowstring material, etc. running along it's back that tenses as the bow is drawn. The native peoples of Alaska and Canada used this bow frequently. It allows inferior materials to be fashioned into a bow. Do a search of Inuit cable bows, for example, and you'll see what I mean about inferior materials. Another method is to do what you've suggested with the bamboo, and that is to use a backing that doesn't cover the entire width of the back. However, I would also classify this type of backing in the following category:
TRAPPED BACKS/BELLIES
Trapping is a fancy word to shaping the cross-section of a the bow limb like a trapezoid. For example, if you're backing is very strong in compression and threatens to overpower your belly material, you would narrow the width of the back, leaving more belly material to withstand the compression load. This also lowers mass, which increases cast and reduces hand shock, etc. Fiberglass bows are commonly trapped, mostly for the latter. Glass is incredibly strong and can take much abuse, so having the back full width is really often overkill. You can also do just the opposite. If you've got a belly that's particularly strong in compression (either naturally or from tempering), you can narrow the width of the belly so that more wood is available on the back to take the tension load, thereby reducing the risk of a splinter.
AESTHETIC/CAMOUFLAGING BACKS
These backings are used to enhance the appearance and/or camouflage the bow in hunting situations and do nothing to increase the safety or shooting performance of the bow. Although this is often accomplished via stains, dyes, and paints, it can also be done by using snake skins, fish skins, various cloth/paper prints, some tree barks, etc.
If your bow is shooting fine now, there's no need to back it. In fact, you can quickly wreck an otherwise good-shooting bow. Although I've built a lot of bows out of red oak and think it can make a fantastic weapon if built carefully, it is not a good candidate for bamboo backing. It's just not strong enough in compression to withstand the crushing load induced by the bamboo. It will either take significant set or fail. Plus, you generally want to add a backing that improves performance (like bamboo) before you've tillered much past 4-6" of tip movement on the long string.
Well, I hope this helps. Leave this bow alone, and get crackin' on another. Best of luck, and use the search function to find examples of all of these types of backings.