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).
I would get a Japanese rod, as mentioned elswhere the Nissin Pro-Spec 2 way in 360 is a GREAT rod to start with. I would avoid the kits, as they contain furled lines which I hate and the rods are not as good.
Here is what I would buy if I were you and wanted to get it all on Amazon ($191 total):
ROD https://www.amazon.com/gp/product/B00B7YHVLO/ref=ox_sc_act_title_4?smid=A1SQEVV7NWZFTS&psc=1
3.0 LEVEL LINE https://www.amazon.com/gp/product/B01GREZB1A/ref=ox_sc_act_title_3?smid=A1KRYSMJ8GAFXD&psc=1
6x TIPPET https://www.amazon.com/gp/product/B015461WVK/ref=ox_sc_act_title_1?smid=A1V2BME88WQ7NI&psc=1
LINE WINDER https://www.amazon.com/gp/product/B01N3PY0H8/ref=ox_sc_act_title_1?smid=A1KRYSMJ8GAFXD&psc=1
FLIES https://www.amazon.com/gp/product/B0776J967V/ref=ox_sc_act_title_1?smid=A89YZQKYPKAF&psc=1
Flies are a bit tricky on Amazon, I don't actually recommend them, but these will not be any better or worse than the ones that come with a kit. If you are in the SF Bay area, PM me and I can set you up with some flies (assuming you fish for trout.)