Calculation of collector resistance for maximum output swing
So far, I have calculated Rb = 0 and Re = 4.3k ohms. But I am unable
to proceed further, for Rc

So far so good.
Next you need to consider the midpoint voltage on the collector that would allow maximum p-p swing for an output signal without clipping top or bottom. Clearly the output could swing as high as +Vcc so that's 50% done but how low can Vc swing?
The simple and straightforward answer (assuming that I didn’t make a stupid error, which of course I did lol) is that Vc can swing no lower than Vb - 0.7 volts else the CB region is forward biased hence the midpoint is half way between +10 volts and +4.3 volts. However, I'd give some margin at the low end to ensure that the base-collector region doesn't saturate (you lose beta when that happens) so I'd say the optimum midpoint of Vc is +7.5 volts.
That means that Rc drops 2.5 volts at 1 mA or conversely, has a resistance of 2.5 kohm.
This calculation assumes that the base resistance Rb is zero ohms (as is usually the case in common-base to prevent miller effects).
Edited section due to mistake in the above calculation.
I made a mistake above by assuming Vb is at 5 volts but of course it’s at 0 volts hence, the lowest collector voltage that can be comfortably reached is 0 volts (or maybe a tad less) but I’m comfortable with 0 volts. The implication is that the Vc midpoint is 5 volts and therefore Rc drops 5 volts and is a value of 5 kohm for 1 mA collector current.