US$1000 speaker challenge...?

In aid of your planning for amps and speakers for use at both home and gigs, it would be good to consider a few issues that distinguish the two cases:

1. Will you want stereo in both cases? Stereo requires more equipment to project into an audience, and attention to assure the L and R contributions to the stereo image blend well in all parts of the room. The near field type monitors that you use at home, with their close placement, blend easily because your ears are the only target for the stereo image. Mono may be a good choice for your gigs, but you may want to hear (monitor) stereo in your ears as you play.

2. How will you hear yourself as you play? When you project your sound into an audience, it will probably sound different to you at your keyboard location, and it will sound different in each room you play.  You will want the ability to EQ your main audience speakers to compensate for the reflections and absorbances that are unique to each particular room. Depending on the EQ applied, these corrected speakers could sound very weird to you, up close at your keyboard. If you are playing solo, this is particularly hard to ignore as you play. You will want a monitor, either IEMs or a small monitor speaker, to hear yourself with the clarity that you are used to at home.

3. What size rooms do you envision for practice and performance? In my opinion, it is unlikely that the same speakers you enjoy up close for practice in your home studio will work for projecting into an audience, but perhaps other forum members have some good ideas for this.

These are normally sold as single speakers, unless specified as a pair. Obviously you will want a pair:

Genelec.
Small rooms are problematic in the bass try without subwoofers.
https://www.genelec.com/8030c

Alternatively to satisfy both very different venues you could use a pair of Behringer "B2031a Truth" & a good sealed subwoofer or two.
In the bedroom you might not need to use the subwoofers as well as they might just give you too much bass.

The Behringer loudspeakers can work well if not squished up too close, or too close to side walls.
If you are practically on top of them they could be a bit overbearing/tiring.  Give them space to breathe I think you should be happy enough with them. So really depends on your room layout whether it's feasible. It's not a large loudspeaker by historic standards but currently people expect miniscule speakers to give them the impression of bass.

So depending on the layout that might be too much loudspeaker, or work well in your small room - if you can give them the space. They should be fine in the larger venue provided you don't need to drown out loud sounds as they're not a PA speaker.  They do have a small wave guide on the treble and a larger than average mid bass driver and will project quite well for a small/medium monitor.

a tiny (relatively quiet) restaurant or cafe will be okay with the 5 inch bass mids too. Anything more challenging in terms of background noise and you would want to look to bigger speakers.

It depends on the available positioning and EQ shelving built in to the active monitoring pair (the subwoofers also having this is a given).

5" drivers can have a pleasing impression of bass. It very much depends on the speaker design and the room acoustics, and your position relative to the speakers.  This is not an easy subject!

At this stage I think you're better off getting a demo and having a listen to some options. Go to some music shops, or Hi-fi shops or both. After that arrange a home demo or three.

Competing engineering compromises..

These small Genelec and Genelec copy (Behringer) have quite small waveguides on the treble - they will be slightly more controlled in directivity compared to a typical box speaker of similar dimensions.
This can slightly improve intelligibility especially as moving further from the loudspeakers. This comes at a cost of a smaller virtual soundstage.

Smaller loudspeakers radiate more of their frequency range in a broader more omnidirectional pattern.  The wider the baffle and wider the drive units the sooner your loudspeakers are beaming sound at the listener (and this has serious pros and cons).

The price you pay for omnidirectionally is a lack of intelligibility when listening from a greater distance. Good omnis can be really excellent in the nearfield though.

Much of what we listen to when we're playing with Pianoteq is really playback of a wet virtual environment,  not a dry piano accurately radiating as if a real piano into a resonant room, listened to from an appropriate distance.