Totally agree that friction loss calculations are a pain! And I know zero about Intex pumps. I do, however, note one thing that will improve your calcs. The friction loss through a 90 is expressed as "equivalent feet of pipe". Engineering toolbox online has your 7.5 figure for equivalent length of pipe in feet, and the calculation is...
7.5/100*3.8 = 0.3 ft of head loss
(where 7.5 is the friction loss for a normal tight radius 90 elbow, expressed in equivalent length of pipe and 3.8 is the friction loss per 100 feet of 1 1/2" PVC pipe at 25 USGPM)
Engineering Toolbox PVC fitting friction loss figures are higher than most other sources. You'll find more like 4 equivalent feet of pipe from other sources.
Don't feel bad about 25 elbows - solar has to go around eaves, perhaps over roof hips, feed and collect from the array, etc. and often has lots of elbows. Reduce all you can, but don't sweat it, and use 45s where it makes sense and looks good (for example around the eaves).
Now the pump - if it's 1/3 HP and is doing other work as well (circulation and filtration), I can't see how it will be enough. It is likely to be a very low-head pump as well. I have a 1/2 HP dedicated pump for solar, going to a heating array on a single story roof (around 10' lift) and it's decently matched with maybe 20% power to spare. The supplier recommended a 1/3 HP high-head. I went 1/2 HP because I want just enough head to do the job, but then highest flow the array could handle once the lift need was met and head loss from lift returned to zero.
In a solar heating system, the pump is selected to have more than enough power to get the water up to the highest point. After the water passes the tipping point, and the return pipes refill, lift is negated and the work done for lift returns to zero (water is taken from the pool's water level and then returned to the same level, so no work is done for this purpose). In rooftop, this is what requires a bit more head from the pump. You also need extra power so you can throttle the flow. If you can't throttle, it's pure luck whether or not your vacuum release valve stays closed when the system is running, or pulls air due to the pull or suction created by the force of water going down the pipes back to the pool. The pressure at the tipping point must be high enough to keep positive pressure on the vacuum release valve, until the system shuts off.
If it were me, I'd consider a dedicated pump or a booster pump. There are experts here that can tell you what's most likely to work for you. I'm new here and there's vastly more knowledgeable people here than me!
Again, if it were me, and I was doing it as cheap as I could, I would try the existing pump by building the supply side of the piping, with maybe 8 or 10 feet of 1/2" or 3/4" at the end to simulate the panels or tubing. I'd use an elbow and point a foot or so of that pipe straight up, open ended. With the 1/3 HP doing all it's other coincident work, if I could shoot the water up a couple feet into the air beyond the end of the pipe, I'd reckon it might work and finish building. If it doesn't work, then I'd buy the dedicated or booster pump. Once the system fills, overcoming friction loss I think will be less than your 20' lift requirements. i.e. my bet is running will be less of a stretch for your 1/3 HP pump than the lift work. Sorry, but the farmer in me can't help at least trying to make do with what I already have. Downside is that even if it works, it will probably be pushing your 1/3 HP to the limit. But maybe deep down you'd way rather burn that one out and then be "forced" to buy a shiny new variable speed pump anyway??? Haha, food for thought. Have fun with your project!