Are you planning on entering an ecosystem? I set up TP-Link Omada in my home. I have a router/firewall, AP controller, switch with PoE ports, and 2 APs.

Firewall: https://www.amazon.com/dp/B08QTXNWZ1?ref=ppx_pop_mob_ap_share

Switch: https://www.amazon.com/dp/B08J9ZQ7HC?ref=ppx_pop_mob_ap_share

APs: https://www.amazon.com/dp/B0781YXFBT?ref=ppx_pop_mob_ap_share

AP controller: https://www.amazon.com/dp/B08SW3GFHH?ref=ppx_pop_mob_ap_share

Good choice! You would be completely fine with just a TP-Link EAP225. You don't need WiFi 6 (your current router is only WiFi 5), and this will keep the price way down for you. This is the exact AP that I put in my parents' detached studio and it has been rock solid.

DEWALT Drill Bit, Spade, 3/8 Inch x 16 Inch (DW1590) https://www.amazon.com/dp/B0001LQYJY

Cat6 https://www.amazon.com/Adoreen-Outdoor-Ethernet-Waterproof-Resistant/dp/B095Y5BMQ5/

Access point

https://www.amazon.com/TP-Link-EAP225-V3-Wireless-Supports/dp/B0781YXFBT/

You are going to to need the Starlink ethernet adapter pretty much no matter what you do.

Ideally:

1. Run ethernet cable from starlink to a switch inside the house
2. Run ethernet cable from the switch in the house to an Access Point (AP) in the house
3. Run an ethernet cable from the switch in the house to an access point in the workshop.

Alternatively

1. Run ethernet from starlink to a point to point wireless bridge in the out building
2. The other side of the wireless bridge will be mounted on the house.
3. Run ethernet from the wireless bridge on the house to a switch in the house
4. Run ethernet cable from the switch in the house to an Access Point (AP) in the house
5. Run ethernet cable from the switch in the house to an Access Point (AP) in the workshop

Edit:

Switch: https://www.amazon.com/Ethernet-Splitter-Optimization-Unmanaged-TL-SG105/dp/B00A128S24/

Wireless bridge:https://www.amazon.com/NanoBeam-NBE-5AC-Gen2-US-airMAX-Dedicated-Management/dp/B07NNWY9Y8

Access Point: https://www.amazon.com/TP-Link-EAP225-V3-Wireless-Supports/dp/B0781YXFBT/

I use this access point in my house. I currently have 2 - one on each side of my ranch-style home.

I also have the Omada infrastructure (TP-Link’s answer to Unifi), so getting this will allow you to scale up to your own ecosystem if you want. If you are interested, I can reply with my entire setup (behind the ISP).

For a simple AP addition, what you propose will work. Just make sure you match the network name (SSID) and pass phrase to your current Wi-Fi, so you have seamless roaming through your home.

Which FIOS Router do you have? The G1100 (MOCA2.0 Bonded) or the G3100 (MOCA2.5 Bonded)?

You can buy the Actiontec MOCA bridge

https://www.amazon.com/Actiontec-Bonded-Ethernet-Adapter-ECB6200S02/dp/B013J7OBUU/ref=sr_1_1_sspa?crid=DG1QKBC8JTZ1&dchild=1&keywords=moca%2Badapter&qid=1593079310&sprefix=moca%2Caps%2C163&sr=8-1-spons&spLa=ZW5jcnlwdGVkUXVhbGlmaWVyPUEzQUtONjdKVkEwMURMJmVuY3J5cHRlZElkPUEwOTYzNDcxMk5ORDdaVzI2RlBIQyZlbmNyeXB0ZWRBZElkPUEwMjAxMjgzMk1UWElHQzRCWlMxMiZ3aWRnZXROYW1lPXNwX2F0ZiZhY3Rpb249Y2xpY2tSZWRpcmVjdCZkb05vdExvZ0NsaWNrPXRydWU&th=1

And then connect it to any Wireless AP you want. Or connect it directly to one or more (with gigabit switch) hardwired devices (Roku, PC, etc.)

TP-Link makes great business class AP's, like this EAP225V3, for example, for $60.

https://www.amazon.com/TP-Link-EAP225-V3-Wireless-Supports/dp/B0781YXFBT/ref=sxts_sxwds-bia-wc-p13n2_0?crid=2ARQ00FOOSS1V&cv_ct_cx=eap225&dchild=1&keywords=eap225&pd_rd_i=B0781YXFBT&pd_rd_r=ecdad363-56fe-432c-a034-6cb0f8934207&pd_rd_w=cg27N&pd_rd_wg=Abidh&pf_rd_p=1da5beeb-8f71-435c-b5c5-3279a6171294&pf_rd_r=SHE3ZJ3Z3J2FXDWSHGWN&psc=1&qid=1593079618&sprefix=eap2%2Caps%2C156&sr=1-2-70f7c15d-07d8-466a-b325-4be35d7258cc

I didn't use a ubiquity router. I used a TPLink ER605 and some wireless access points.