Sure, this is exactly what I do; backpack the Sierra.
First off, you should know that for stillwaters (lakes) Tenkara is not as good as Western rig rods. They don't have the same reach and they aren't as versatile in the wind.
They are lighter though, which you understand is a key component of backpacking. I can fish on my trips for well less than 4 ounces.
Beyond buying pack rods, I have also made the mistake of trying to buy a longer rod to increase stillwater reach, -don't bother. The difference between your bog-standard 12' rod and a 15' rod makes no real difference.
$150 is a reasonable starting budget, but it will send you into a lower class of rod than necessary if that is a bright line. If you can up that by $20 or so you can get a good Japanese rod + the stuff you need.
Rod Recommendations:
$10 6x Tippet
$? Flies
If the $150 is an absolute max, you are probably best off with a Dragontail rod. I have heard that they are the best at the price point, but have not fished any of them.
Beyond my general advice to stick at the 12' length (360cm/3.6m), I would say get whichever Dragontail sounds good and other folks here recommend.
I meant it too, if you are in the Bay Area hit me up. I can put a bunch of rods in your hand to get a feel for what you like. Can hook you up with some free flies too (I tie).
If you are a backpacker and want a specifically short rod (< 24") then get the Diawa Keiryu-X 360cm (not available on amazon, but only about $110)
If standard Tenkara lengths (approx. 24" collapsed) are fine then the best rod for your money is a Nissin Pro-Spec 360cm/310cm zoom rod: https://www.amazon.com/dp/B00B7YHVLY/?coliid=I2YPR8C72BS0VX&colid=FTRENIC5N1AI&psc=0&ref_=lv_ov_lig_dp_it