but if that isn't possible / tweeters that small aren't worth having
The size of a driver (be it a tweeter, mid, or woofer - though not so much subwoofers) should be matched to the system. Smaller drivers weigh less and can typically use a relatively larger motor because the smaller cone has less flex. This means they can play higher in frequency. Also, with typical dynamic speakers (especially cones) there is a property known as "beaming" where essentially larger cones have less angular dispersion than smaller cones. If you had speakers in your living room with a relatively large 2" tweeter and you sat directly in front of your speakers, you would be okay. But if you had friends over and they sit way off to the side, they will hear a lot less treble than you. But if you were using a nice 3/4" dome or ring radiator tweeter (like the excellent Vifa XT19 19mm ring radiator my DIY front speakers use) your friend will get sound that is much closer to what you hear (though still not perfect).
The upside to larger drivers is that they can typically handle more power, have more excursion capability (Xmax / Xmech), and can play lower in frequency range. Edit: Oh, and as previously explained, they are often more sensitive, though not always. Depends on the motor design.
There is also a downside to larger angular dispersion: Your room is likely to cause more reflections which can cause phase issues and the "combing" effect. There are ways to mitigate this but not that easily, so for a budget system it's something you'll just have to deal with. It shouldn't be a super major issue, though.
Since your woofer/midrange is fairly small, you should be able to get away with a small tweeter like 3/4". If you had a 6.5" or especially 8" woofer (without a midrange i.e. staying with a 2-way system instead of a 3-way) then you would be more likely to need a larger tweeter to match, because a larger woofer will roll off (and start beaming) sooner than yours, so ideally the tweeter would be able to cross over at a lower frequency.
As a side-note, I prefer 3-ways for high-end home speakers, but there's always a tradeoff - the crossover. They become more complex and expensive and you really want to avoid having to using L-pads to match the volumes in a passive crossover. An active crossover isn't so much of an issue if you have enough amp channels (in your case you don't but this is more general speaker knowledge anyway).
If you do use a separate tweeter with that woofer, I'd clip the tweeter lead for the coax. It's probably just using a 1st-order filter consisting of a capacitor. Don't try to remove the cone, though, or you will probably destroy the speaker (and if you didn't, you'd need to buy a dustcap for it to cover the hole).