From what I understand, any timer switch installed would have to be three-way capable as well, which not many are. I'm not sure about the two switches RWD linked, but from the general searches I'm finding online? As far as I can tell, anyone else in this predicament (timer switch on a three way switch circuit) gets recommended an Intermatic ST01 switch, which runs about $55 currently.
This kinda feels like throwing good money after bad, and you'll understand why here shortly.
I don't know how the lighting is wired in the basement, but I'm picturing hard-wired edison socket(s) in the ceiling of a basement either unfinished or partially finished. If this isn't the case, adapt the following accordingly to your lighting configuration. I'm assuming that it's not designed to be a living space given the description, and the lighting is appropriately minimalist in execution.
If it's just a single (or even a couple edison sockets)? Seriously, just stick a 40W LED equivalent in and let it stay on. At 3W a bulb, even at outrageous power prices like $1/kWh (which, even Hawaii's peak pricing is still under 50¢/kWh), that single bulb would only cost $1.10 to run
for the year.
If there's multiple sockets and you still want most of the lights off unless you're deliberately down there? Just install
cheap pull-chain switches for the other socket(s)/lights. But even then, at the national average of 15¢/kWh with a steep compounding 10% inflation, it's still going to be 15+ years before you see an ROI on each of those $5.46 sockets versus just leaving that 3W bulb on 24/7/365.
(Later edit: I didn't think to include replacement bulb cost in this calculation. Factoring a 2.5 year average lifespan for 100% on and $2.50 a bulb, you could break even between 5-10 years, which actually isn't a bad ROI timespan, even if we're talking about a $5.50 expenditure... excuse the labor.) I can't imagine how long the ROI would be with a $55 timer switch, or even a $25 switch... never mind the question on whether those timer switches would even last long enough to see a net positive ROI versus low wattage bulbs before needing to be replaced given the quality of most electronics these days.
Sometimes, the cheapest and easiest solution is to just lean into the bad habit in the least wasteful way possible if the habit can't be broken. A couple low wattage LED bulbs left on can actually be less wasteful than trying to rewire an entire basement just to automatically turn the lights off.
