Content ScriptからBackground Script内の関数を直接呼びだす(ように見せる) ref: https://qiita.com/tkatochin/items/30e077f7a8f94d39d714

backgroundCaller.js

/**
 * バックグラウンドにRPCする呼び出し側の実装（manifestのcontent_scriptsに記述して配備すること）
 */
'use strict';
(function(){

    function createAsyncRpcFunction(methodName) {
        return async function() {
            const req = {
                method: methodName,
                args: [].slice.apply(arguments)
            };
            return new Promise((resolve, reject)=>{
                chrome.runtime.sendMessage(req, function(res){
                    if (res && ["OK","ERROR"].includes(res.status)) {
                        if (res.status == "OK") {
                            if (!res.data) {
                                res.data = undefined;
                            }
                            try {
                                resolve(res.data);
                            } catch(e) {
                                console.error(`then-callback call failure!! ${e}`);
                                reject(e);
                            }
                            return;
                        } else {
                            console.error(`RPC ${methodName} result ERROR!! ${JSON.stringify(res)}`);
                            reject(res);
                        }
                    } else {
                        console.error(`RPC return value of ${methodName} is no reply or unknown status!! ${res}`);
                        reject(res);
                    }
                });
            });
        }
    }

    /**
     * RPCの仕掛け。
     */
    window.bg = new Proxy({}, {
            get: function(target, methodName){
                if (methodName in target) {
                    return target[methodName];
                }
                return target[methodName] = createAsyncRpcFunction(methodName);
            }
        });

})();

backgroundListener.js

/**
 * RPCされる側の実装（manifestのbackground/scriptsに記述して配備すること）
 */
'use strict';
window.bg = window.bg || {};
(function(){
    chrome.runtime.onMessage.addListener(function(req, sender, sendResponse) {
        if (req.method in bg && bg[req.method] instanceof Function) {
            let func = bg[req.method];
            try {
                var result = func.apply(func, req.args || []);
                if (!(result instanceof Promise)) {
                    let trueResult = result;
                    result = new Promise((resolve,reject)=>{
                        try {
                            resolve(trueResult);
                        } catch(e) {
                            reject(e);
                        }
                    });
                }
                result.then(result=>sendResponse({
                        status: "OK",
                        data: result
                    })
                ).catch(e=>sendResponse({
                        status: "ERROR",
                        error: e.toString()
                    })
                );
            } catch(e) {
                sendResponse({
                    status: "ERROR",
                    error: e.toString()
                });
            }
        } else {
            sendResponse({
                status: "ERROR",
                error: `method not found - ${req.method}`
            });
        }
        return true;
    });
})();

file0.txt

var result = await bg.myMethod(param1, param2);

file1.txt

bg.myMethod = function(param1, param2) {
    var result = ...;
    ...
    return result;
}

file2.txt

bg.myMethod = function() {
    return new Promise((resolve, reject)=>{
        needCallbackLogic(function (result){
            resolve(result);
        }, function(error){
            reject(error);
        });
    });
}

file3.txt

bg.myMethod = async function() {
    return await needCallbackLogic();
}

file4.txt

function createAsyncRpcFunction(methodName) {
    return async function() {
        const req = {
            method: methodName,
            args: [].slice.apply(arguments)
        };
        return new Promise((resolve, reject)=>{
            chrome.runtime.sendMessage(req, function(res){
                if (res && ["OK","ERROR"].includes(res.status)) {
                    if (res.status == "OK") {
                        if (!res.data) {
                            res.data = undefined;
                        }
                        try {
                            resolve(res.data);
                        } catch(e) {
                            console.error(`then-callback call failure!! ${e}`);
                            reject(e);
                        }
                        return;
                    } else {
                        console.error(`RPC ${methodName} result ERROR!! ${JSON.stringify(res)}`);
                        reject(res);
                    }
                } else {
                    console.error(`RPC return value of ${methodName} is no reply or unknown status!! ${res}`);
                    reject(res);
                }
            });
        });
    }
}

/**
 * RPCの仕掛け。
 */
window.bg = new Proxy({}, {
        get: function(target, methodName){
            if (methodName in target) {
                return target[methodName];
            }
            return target[methodName] = createAsyncRpcFunction(methodName);
        }
    });

file5.txt

const constants = new Proxy({},{get:function(target,name){return chrome.i18n.getMessage(name);}});

file6.txt

const constants = new Proxy({},{get:function(target,name){return chrome.i18n.getMessage(name);}});