The any 9000BTU (cooling mode rating) mini-split heat put would heat & cool your basement, and would be able deliver something like 11-12000BTU/hr at full compressor & blower speed at your outside design temp.
On the noise levels, note that most refrigerators that put out noise higher than 40dbA. I set up my mother's place (western WA, design temp of ~20F) to heat her whole house with a 1.5 ton Mitsubishi (good for output of ~21KBTU/hr in heating mode down to about +5F, falling to~15kbtu/hr @ -13F). It's louder than her refrigerator only at the highest blower speed, but she programmed it for low-speed only, even though that limits the maximum heat output. She may kick it up during a cold snap, but she hasn't had to do it yet. (It was installed in late February this year, IIRC.) Odds are pretty good that your basement mini-split would never have to come off the lowest blower speed, but may modulate a bit in compressor speed.
Any of the models I suggested are inverter drive with fully variable compressor & blower speeds, as are the better LGs. Some cheaper interior cassettes/heads are 2-speed AC motors and not quite as efficient or quiet, but they're still not bad. Any mini-split with an HSPF rating better than 8.5 will have a fully variable speed inverter drive compressor though, even if they might use cheaper blower motors & controls on the cassette.
SFAIK Daikin doesn't make a 2-head Quaternity to take advantage of the more sophisticated dehumidification mode. If that's a compelling feature for you it may mean buying two separate units. If the basement and the addition are close to one another it may make sense to do one of those. If separated by a longer distance than 10meters/35 feet from where you set up the outside unit you can run into issues with maximum lengths on the refrigeration lines and control cables, which would drive you toward two complete units.
An off-the top of my head, a short list of pretty-good ~1.5 ton dual heads are
These all run ~18 SEER, and in the low-9s for HSPF.
With most 1.5 ton duals it's easy to mix 'n match heads: 9K + 9K, 7K+12K, 7K+9K, or 7K+ 7K (derating total output down to 14KBTU/hr which might be right for your situation unless you anticipate a bigger cooling load than 7KBTU/hr for the addition, which may be the case.) The hardware cost for the outdoor unit would run ~$2.5-3K, then about $350-500/head (x2), plus installation, so it would likely hit $5-6K, installed.
A pair of 3/4 ton LGs like the LS090HYV may cost only slightly more (or maybe even less than a dual head Daikin), and comes with more favorable efficiency numbers (SEER28/HSPF12), but it's hard to say. With most of these the cassettes are powered by the outdoor units, which limits the amount of AC wiring necessary for installation. Some installation issues double when you install two outdoor units, but it can sometimes be simpler to go with two.
It's a moving target and a competitive market, since small mini-splits hold a lion's share of the Asian space heating. You can even buy a Chigo in the US, if you thing you will be feeling nostalgic about Afghanistan, but SFAIK they don't have any super-efficient models yet like the big Japanese & Korean manufacturers do. The US is a late-comer to the technology, and with larger more sprawling & doored-off houses ducted systems are still the paradigm, but as awareness grows, so does the mini-split market.