Taking in to account your desire to charge your laptop, the largest battery I could find within your budget and from what appears to be a reputable seller would be the BiXPower BP-160.
Slightly over $200 at $209+shipping.
Capacity of 153Wh, equivalent to 42,500mAh@3.6V. This is likely 2-3x the capacity of your laptop's internal battery.
Not overly small (...
Mini-Box OpenUPS does the job but is much too expensive at $150 (Amazon Germany - $120 directly from the manufacturer). I won't accept this answer because I think something much cheaper should exist.
It does everything asked for in the question, plus balancing up to 6S and some other battery chemistries too and output is not just in the right range, but ...
I use an EasyAcc PB12000 (12,000 mAh) to power my Raspberry Pi. Given how long the Pi will run on a single charge, it does appear to reach its claimed capacity.
EasyAcc doesn't appear to sell that model any longer, but a pair of these 15000 mAh batteries should meet your criteria:
Total capacity: 30,000 mAh
Total cost: $54
Total weight: 1.25 pounds
I found This Runsy iPhone 7 case It looks like it uses a lightning passthough, so you won't need to carry a pesky micro usb cable around. The only limitation is that it does not work with headphones with lightning connectors.
Overall, it has a 4 out of 5 star review. Let me know if this fits the bill.
After a quick search, I found these battery devices:
XTORM 18W POWER BANK APOLLO 15.000
XTORM 45W POWER BANK ROVER 20.000
XTORM POWER BANK APOLLO 15.000 (LIGHTNING)
XTORM POWER BANK ROVER 20.000 (LIGHTNING)
AIDEAZ Wireless Power Bank 20000mAh Portable Charger
There are others mophie powerstation USB-C 3XL Battery with one USB-C as output, and another one ...
So there's still the unknown of why a Dell 45W USB-C charger did not work on a Dell machine. Given your measured power draw, it might be that your machine doesn't support that charger without some firmware update or just doesn't support charging via USB-C. Maybe try some other USB-C laptop chargers (45W+) you can get a hold to see if they work. For this ...
If your AC adapter is 200W, your situation is similar to mine. I have a 120W laptop and 230W gaming laptop, and here are the solutions that I found.
1) ChargeTech 54000 mAh at $399 - Top-of-the-line solution that will power any device up to 250W. Still the price is quite steep....
This uses a mains AC output, and so I'm unsure about how it works for ...
Check out openUPS2 or NUC-UPS from mini-box.com, they even have hard drive mounting footprint for easy integration with case. They also send ON/OFF pulse to the motherboard via 2 wire or USB for graceful shutdown.
Every battery case I've seen (for phones with non-removable batteries) occupies the port. Meaning that you can't use it for other interfaces. So that seems like an unlikely solution (not to say it isn't possible or doesn't exist). I realize this doesn't exactly match the criteria presented by the question, but please bear with me. What if you used a USB C ...
I did some research and found that most power banks (and laptop batteries) use 18650 Li-Ion batteries (cells).
So finally I bought a 18650 battery case (search for "18650 power bank case") which I'm using with a 18650 cell from a disassembled old laptop battery. Currently this one:
Product link and image source: 2600mAh 5V 1A USB Power Bank Case Charger ...