Water Filter Hack

My kitchen has pull down faucet like this.

The problem with this is that I can’t attach a counter top filter something similar to this
as it needs standard faucet to fit the diverter.

I asked my apartment management if I could get a faucet of my own. They denied the request and said only they can make any modification and that I have to pay for it (could cost $100 - 150) and I cannot take away the faucet as the lease says any modification remains with the apartment.

That got me thinking if there are any water tank based counter top filters. I found couple of them but costs upward of $400. My plan is to either

  1. buy a small water motor pump that will pump water from a 5 gallon pet carboy and attach filter piping to it.
  2. Attach the filter to a carboy with spigot placed at a height.

I request your opinion and suggestions if this works or if there are any other way to solve this problem.

If the options works
With first option, I need help with the water pump specification (like wattage or pressure) and with second option, where can I find mods to fit the filter pipe to carboy spigot (or may be the drain hole).

I suggest these guys, they are the gold standard!

Then nephros is the platinum.