I assume the hatch to attic in the closet means you have access for ease of mounting. And that the hatch also makes it so you can't mount in center.

How about some LED linear lights? They mimic the old style fluorescent, but have much better look and last a long time. With LED you can also choose a lower color temperature than traditional fluorescents. I prefer 2700k-3000k indoors, but think you will want ~3000k in a closet, maybe slightly higher color temperature. Also, you can try to get better CRI so that your clothes, etc look better when putting things together. Maybe you can use one on each side of the hatch?

How about something like this:

https://www.lightingdirect.com/lithonia-lighting-fmfl-30840-caml-oa-13-38-flush-mount-3000k-led-ceiling-fixture/p2761784

A listing of LED linear lighting fixtures under $200 can be found here:

https://www.lightingdirect.com/ceiling-fixtures/c1206?f59763=under%20%24199&f60951=led%20fixtures&f64255=linear&r=48&s=SCORE&p=1&categoryId=1206

I have used Lighting Direct recently, like them, and most importantly am familiar with the web site. I don't work for them!

Anyhow, my advice is to not shy away from linear lights because they have improved a lot.

-Adam

Thanks Adam! Here is a picture; the attic hatch is pretty much in the middle of the room; the centre of the light is positioned ~13cm/5" from the edge of the trim around the attic hatch. I'll check out the LED linear options - we're doing something similar, but wall-mounted, over the bathroom vanities...

You might be able to get a linear light that covers the existing hole and still connects to it. But 5" is not much room! I'm surprised it was mounted so close.

If you planned to get tracks in there, I assume you are having someone do the work. Electricians aren't cheap, but you could get someone to move over the junction box and put another one on the other side of the hatch. Then you could add the linear lights, or just have them do it. There are codes that must be followed (as you know from track advice), and you need someone that knows what they are doing. But apart from putting a buncy of recessec cans in there, 2 linear lights on either side of the hatch seem like they will work. The nice thing is you have easy access, so time in attic is minimized.