Yes, that setup can work, but the outdoor unit is the limit, not the sum of the indoor units.

With a multi-split, the indoor units can usually add up to more than the outdoor unit — manufacturers publish a combination table and a maximum ratio, often around 120–130%, so check the exact model you are buying. A 27,000 BTU outdoor unit with 7,000 + 9,000 + 9,000 BTU indoors is a common pairing. If you run only the living room and one bedroom together, as you plan, you are well within that capacity on most days. If all three run at once on the hottest or coldest day, each room will get less than its full rated output, so they will cool or heat more slowly.

For your room sizes, in a new, well-insulated apartment, 9,000 BTU for the 200 sq ft living room is usually enough, and 7,000 BTU will usually cover the 100 sq ft bedroom. A 12,000 BTU unit in the living room would only make sense if there is a lot of glass facing south, high ceilings, or a heat-gain problem — otherwise it can short-cycle and leave the room feeling clammy in cooling mode. The 5th floor, south-facing aspect does add afternoon heat, so that is worth mentioning to the installer.

Do not size this from square footage alone. Ask the installer for a proper room-by-room load calculation, and to confirm the outdoor unit, indoor combination, pipe lengths, electrical supply, and condensate drainage all match the manufacturer's table for that exact system. That is also the point to check whether the existing outdoor holder, wiring, and breaker can take the larger unit, before you buy it.

I run Ocala Pro AC Repair here in Ocala, FL — we see this a lot in Florida heat, and oversizing is one of the most common mistakes with mini-splits. Get the combination confirmed in writing, and you should be fine with two rooms running together.

Your sizing instinct is right, but the limit to check is the outdoor unit's rated indoor combination, not the square footage. Pull the exact outdoor model's spec sheet and look for its allowed indoor capacity ratio and combination table, because a 27,000 BTU unit with 7,000 + 9,000 + 9,000 BTU heads is a common pairing only on some models. Before you buy the bigger unit, measure the existing bracket and wall anchors, since mounting capacity often decides the question more than cooling capacity does. Ask the installer for the maximum line length and refrigerant charge for each indoor head, the breaker and wire size from the nameplate, and a written room-by-room load estimate for the south-facing living room on the 5th floor.

The OP's post is two years old. Hopefully they resolved their issue a long time ago.