X-ray fluorescence Non-Destructive Techniques are helpful. Special systems were designed to investigate the chemical composition of valuable archaeological artifacts and statues. Alternatively, a tiny chip (or very small amount of filings from the vessel which essentially would not affect its shape) can be investigated by Energy-Dispersive (EDX) or Wavelength-Dispersive (WDX) x-ray spectroscopy facility available in Scanning Electron Microscopes (SEM) or STEM systems. For accurate results, however, a calibration curve is needed. The fundamental method based on theoretical models can give a good answer, but not so accurate. Of course, extracting the metals is a destructive process. For that, I think a metallurgist can give a good advice.
X-ray fluorescence Non-Destructive Techniques are helpful. Special systems were designed to investigate the chemical composition of valuable archaeological artifacts and statues. Alternatively, a tiny chip (or very small amount of filings from the vessel which essentially would not affect its shape) can be investigated by Energy-Dispersive (EDX) or Wavelength-Dispersive (WDX) x-ray spectroscopy facility available in Scanning Electron Microscopes (SEM) or STEM systems. For accurate results, however, a calibration curve is needed. The fundamental method based on theoretical models can give a good answer, but not so accurate. Of course, extracting the metals is a destructive process. For that, I think a metallurgist can give a good advice.
To analyse for metals nondestructively the use of XRF (X-ray fluorescence) would likely be easiest. This analysis can be made with minimal preparation but does require specialized equipment. Metal extraction would require normal industrial processes and would result in complete destruction of the item.
I suppose you do not want to damage the sculpture, so my opinion is to use a hand-held XRF instrument. Gold and silver are fairly heavy elements, and are easily measured by XRF.
Don't forget that nondestructive methods like XRF (or any analysis - EDS, AAS - of tiny amount of sample from surface) are all techniques which are characterizing surface only. Not only sculptures but any old objects can be gilded/plated and under this layer material with completely different composition can be found (and corrosion or patina can affect surface techniques results aswell). If the sculpture is hollow, you can try to get some small piece of matter from inside. If it is possible and sculture is not too big, very useful can be hydrostatic weighing (applications of Archimedes rule). This can confirm/disprove very easily and cheaply the composition found on surface by other techniques (ofcourse too complicated alloys is hard to determine, but for Ag/Cu, Cu/Au or Ag/Au alloy it si quite easy). For example: if XRF will find 80% of Au and the density of object will be e.g. 11.2 g.cm-3 (and no cavities are present) than it is clear that object is not composed from material with 80% of gold (inside can be e.g. silver or lead). Very useful for binary and ternary Cu, Ag, Au alloy is article from Kraut and Stern (first link). Once I have nondestructively analyzed some gold medal (second link) which has nearly 100% of gold on surface but on some scuffed places only 70% Au, 20% Cu and 10% Ag was found with EDS and than confirmed with hydrostatic weighing (possible compositions of ternary alloy with known density is expresed by the red line).
For quantitative analysis of archaeological artefacts there are set of methods applicable such as for example Raman microscopy; LA-ICPMS, MALDI-MS, XRF which at first approximation are regarded as non-destructive methods, excluding ICP-MS. However Raman microscopy is inapplicable to quantitation of alloys. For ICP-MS, which can give you meaningful information exactly for metals, you need mass of your artefact about 30 mg, which can be taken by steel drill. The size of the drill is 0.1-0.3 mm. The sample is dissolved in 4mL aqua regia. You need however minimum 3 replicated samples from min 5 independent sample areas to achieve meaningful from an analytical chemistry point of view data. Despite this the LOD is 0.01-0.003 % for the elements discussed. So that you can even operate at lower masses of the artefact. This depends on the analyte component concentrations. Solid-state MALDI-MS operates with laser ionization desorption approach and is described also generally as non-destructive. This method allows direct assay. Please pay attention to the attachment where is shown what shall be the damage of an object's surface upon laser ablation. The area of the laser shot is 10-20 microm. However, preliminary it is difficult to evaluate are you shall obtain good ionization efficiency under direct analysis without matrix components exactly for your sample content, which to allow quantitation. XRF is used for analysis of alloys, too. It, however, also involves laser ablation so that the comment above is valid to this method as well as, generally speaking about destructive methods. However accounting for the fact that XRF and MS-based methods operate with low concentration LODs as pointed out above effectively you will not cause to a destruction of your old artefact excluding the mentioned above micro-level damages of the surface. The next question, which you should clarify preliminary is are you need gradient analysis in bulk direction or only 2D surface screening?!