QT-Primos 150 - My (Mazak) dealer at the time said YES it is possible to import an QT-Primos 150 from Singapore.
Not sure if that's still the case and how long you'd have to wait / lead times ?
main spindle integrated DD type spindle - very accurate for concentricity and roundness , maybe overkill ? but very compact machine.).
Nakamura: AS-200 very solid machine. (not sure if it's overkill for what you need + Budget $$$ ?).
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Hardinge GSi series (GSi 150 ) - has a small footprint that would fit your space and cutting requirements (perhaps) ? It's not a gang tool lathe but built like a compact more conventional turning center - wedge -ish (one piece) type casting + hydraulic tail stock if you want it.
Affordable/more affordable plain vanilla offering - chip tray to "Muck out" ,
Not sure of your production requirement vs. emptying / raking out a chip tray every now and again ?
GSi Series | Hardinge
https://www.hardinge.com/wp-content/uploads/GS150i_1363-1A.pdf
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Hurco Tm6i might be worth a gander - fits without chip lift up conveyor possibly ?
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Any issues on ceiling height ? As for your 5 axis / 5 axis mill turn requirement most machines - (vertical mills) - really - have to allow for 105" 110" +++ (Height). Some of the Brother machine excel at being garage (door) height friendly but the M300X3 are taller (as well 103 ish " ). but the M200X is closer to 100" tall. [ I hear you on depth ].