Our Florida was not put in until after the summer so we have not yet added a split A/C.
Waiting to see how it will be cooled by our A/C we have now but thinking a split will be needed.

We installed remote controlled awnings. The height and extension can both be adjusted to suit. It has made a HUGE difference in our lanai, which we use all times of day. We use our awnings almost every afternoon. The only time we don't is when it's very windy or (obviously) raining. We installed two - one also shades our bedroom. They were not an inexpensive purchase, (several thousand). However, the comfort they provide, as well as facilitating much increased use of our lanai, has for us, justified the expense.
Contact:

Contact: Sun Shades of Central Florida ph 407-935-9115

There are screens that block more than 90% of sun rays; however we still couldn’t sit out there when it was raining or too cold. We eventually “partially” enclosed or lanai, installing stacking siding doors. Now we have full air when we want, but can keep the doors closed and still enjoy the lanai 100% of the time. Best investment in our opinion.

I have an east faceing lanai. I love it. Sun in the am and cool in the pm.

I've had both and prefer not having the low glaring sun in my eyes every morning, I'm not a big fan of shades and we use our Lanai every morning but rarely in late afternoon as there is always something else to do so West is Best at least for us.

We would highly recommend outdoor solar shades for a west facing lanai. It made a huge difference for us. We got ours from Budget Blinds and they did a great job. We loved the selection of window coverings for both inside and outside our Home!

Cool breeze roll screens,retractable doors,privacy-hurricane shutters and awning
 

Cool Breeze roll screens, retractable screen doors, interior motorized shades,privacy-hurricane shutters and awning.

We manufacture our products in Odessa, Fl and have the Longest warranty,quality and service in the industry.

888-896-8935
coolbreezerollscreens@Yahoo.com

I think using your home air conditioner to cool lanai is less efficient than getting a split. We use ours a lot and really notice little in the way of a higher utility bill.

I was going to share my story but you already did. I have a west facing porch on a lake in NY...sunsets are great, but it can get brutal. However, we have found ways to deal with it, and it has its pluses. Which is what I want to share with the OP...you'll find ways to enjoy the pluses, and to mitigate the minuses. It will be grand.
Enjoy!

We cut down the trees because we love the sun and don't want the trees to block the solar panels that heat our pool.

On a monthly basis your probably right but when you factor in the cost of buying the split would disagree.

We have a Gardenia with a west facing lanai. Like you, we had heard all the reasons why this was a terrible idea. We often think about everything we heard each morning when we have breakfast facing the rising sun in the east and in the evenings when we often just sit and enjoy the beautiful sunsets in the west. Yes, summers in Florida can be brutal - no matter which way your lanai faces. Screens, awnings and other devices can provide some relief if you find you truly want them (we do have pull down screens we use when we must). Enclosing the lanai is a different alternative. We didn't need additional living space nor a replication of our living or dining rooms. We prefer the open air and sense of being outside that the screened lanai provides.

A number of folks have provided the best advice. Live with your house and your lanai. The experience will let you know what is going to work best for you. It might be any of the suggestions already mentioned. Enjoy your home and welcome to TV.

West Facing Lanai:
 

We live in Monarch Grove and have a lanai with birdcage that faces west. We have lived here since October and have now ordered screens from Lowes. We are going to purchase a section at a time to see what we really need. Also, thinking of purchasing a canvas cover for the birdcage.

We have a West facing lanai and love it. The morning breakfasts are wonderful and the evening sunset are fantastic. We were also faced with what "should we do about the sun in the evening" and after some thought decided to go with curtains. We found curtains on Amazon that are mold resistant and have guide holes on both the top and bottom. You need the guide holes on the bottom to stop the curtains from blowing into the lanai. We then had a handy-man run a steel guide wire across the top and bottom of the lanai opening where the curtains were to be placed. A curtain was hung on the left side of the lanai able to run about 1/2 way into the middle and another one hung on the right side able to run about 1/2 way. The logic I used was: Its a lot cheaper solution then an electric shade running from the top to bottom of the lanai window and if we didn't like it we didn't have a lot to loose compared to the electric shade solution. When we want to block the sun typically we only need to move one of the curtains over some and it leave the rest open for a nice view. Out lanai is large and we have 2 sections with 2 curtains on each section. It is very rare for us to have to move even 1 curtain over to the middle and I can't remember ever having to close 2 of the curtains and never the 4. The curtains move fairly easy on the guide wires but clearly not as easy a pressing a button. This solution works for us but as most people have recommended > use the Lanai in the spring/summer/ a little before you make a decision. Good luck.