Band pass filter for RF signal meter circuit
Because of the narrow band should I go for a LC filter circuit?
Yes because LC tends to have higher Q (theoretically). You must pay attention to L:C ratio because it determines the steepness of the rolloff curve.
Small current is associated with large Henry value. This might affect your decision.
There are bandpass filters which make use of capacitors only. They might be suitable, if adjusted as necessary. You might get narrow bandwidth as in your spec.
The quantities you indicated are not enough to specify a bandpass filter. Rejection bands should also be specified, as well as pass-band ripple which will indicate filter type.
What is the RF power ? What is operation temperature and limits? Connnectors?
A BPF at 1700 MHz is typically realised as coaxial coupled-cavity filter, in this case, as BW is narrow, several cavities will be needed.
Your question indicates that you never designed an UHF filter. Read a good textbook or take a course.
The best book I can recommend is Matthaei, Young, Jones : Microwave filters and couplers, issued in 1980 approx.
An LC filter at this frequency AND bandwidth it will not work properly mainly due to the low values of components, which will be in the range of fF and pH.
Cavity or a SAW can give appropriate performance. For SAW filters check, Murata, TDK, Taiyo Yuden, Tai-saw