Help about in instruction of LVDT
Use your knowledge of inductor calculators with sleeves and stainless steel cores and follow the best-in class impedance ratios of similar LVDT's.
e.g. 280 Ohms primary at 5kHz, 1000 ohms secondary for +/-100mm, yet you only want +/-50mm.
The coil range must extend beyond the core to minimize changes on mutual inductance effects on loaded output. 
Choose a frequency that is at least(!) 10x higher than the desired bandwidth for ripple rejection or define your ripple attenuation from sensitivity to small steps in position.
ref:
https://singer-instruments.com/introduction-to-lvdt/
https://singer-instruments.com/wp-content/uploads/2019/10/2_DC_LVDT_S-D_and_M-D_Series-1.pdf
Then follow their construction advice and but I don't know how to estimate the mutual coupling effects of a transformer using a (non magnetic stainless steel core. An aluminum core is non-magnetic, yet has mutual coupling from the eddy currents at 50kHz might be an option. (or not)
It all depends on your goal for source and load impedance as well with changes in mutual coupling, complex permeability, skin effects and eddy current effects. .
For lower mass sleeve cores might be possible or just smaller diameter and the same for inner stainless steel coil sleeves. The steel sleeve and centre core must be smooth and co-linear as possible with perhaps some lubricant (?) and must be precise with the smallest air gap yet have to prevent friction at all costs so that temperature effects does not warp it or enlarge it more than the coil form.