<apex:page controller="DynamicCustomLabel" contentType="application/json" showHeader="false" sidebar="false"
    language="{!lang}" standardStylesheets="false">{!$Label[name]}</apex:page>

public class DynamicCustomLabel {
    public String lang {get;set;}
    public String name {get;set;}

    public  DynamicCustomLabel(){
       Map<String, String> reqParams = ApexPages.currentPage().getParameters();
       lang = reqParams.get('lang');
       name = reqParams.get('name');
    }
}

@RestResource(urlMapping='/customlabel')
global with sharing class DynamicCustomLabelResource  {

    @HttpGet
    global static String getLabel() {
        RestRequest req = RestContext.request;
        String name = req.params.get('name');
        String lang = req.params.get('lang');
        try{
            return getLabel(name, lang);
        }catch(Exception e){
            RestContext.response.statusCode = 404;
            return '';
        }
    }

    public static  String getLabel(String labelName, String language){
        Pagereference r = Page.DynamicCustomLabel;
        r.getParameters().put('lang', language);
        r.getParameters().put('name', labelName);
        String labelValue = r.getContent().toString();
        return labelValue;
    }
}