Firebase Cloud Functions - Upload a File object to Cloud Storage

In my function I create a pdf file using the pdfkit library:

exports.getPdfUrl = functions.https.onRequest(async (req, res) => {
    const bucket = admin
        .storage()
        .bucket();
    //target file name
    const file = bucket
        .file('test.pdf');

    //creating the pdf file
    await new Promise<void>((resolve, reject) => {
        let stream = file.createWriteStream({
            resumable: false,
            contentType: "application/pdf",
        });
        stream.on("finish", () => resolve());
        stream.on("error", (e) => reject(e));
        const doc = new PDFDocument({ size: "A4", margin: 50 });
        doc.text("some text", 50, 50);
        doc.pipe(stream);
        doc.end();
    });
    ...

This works so far and creates a proper pdf file.

Now I want to upload the created file to storage and return a url.

I can't to this, because the getSignedUrl expires maximum is 7 days (I need a permanent url) :

    const url = await file.getSignedUrl({
        version: "v4",
        action: "read",
        expires: Date.now() + 7 * 24 * 60 * 60 * 1000,
    });

I can't do this, because the first argument of bucket.upload() is the path to the file, which I don't have, because I only have the File object:

    const [uploadedFile] = await bucket.upload('test.pdf', {
        destination: 'testNew.pdf',
        metadata: {
            contentType: 'application/pdf'
        }
    });
    await uploadedFile.makePublic();
    const publicUrl = `https://storage.googleapis.com/${bucket.name}/${encodeURIComponent('testNew.pdf')}`;

    const tmpFilePath = path.join(os.tmpdir(), fileName);
    // Create a read stream from the Cloud Storage file
    const readStream = file.createReadStream();

    // Create a write stream to the local temporary file
    const writeStream = fs.createWriteStream(tmpFilePath);

    // Pipe the read stream to the write stream
    readStream.pipe(writeStream);

    // Listen for the finish event on the write stream
    await new Promise((resolve, reject) => {
        writeStream.on("finish", resolve);
        writeStream.on("error", reject);
    });

This will create a the file in the main directory of the storage bucket.

From there it can be saved to other locations:

    const [uploadedFile] = await bucket.upload(tmpFilePath, {
        destination: `someOtherPath/${fileName}`,
        metadata: {
            contentType: "application/pdf",
        },
    });
    await uploadedFile.makePublic();
    const url = `https://storage.googleapis.com/${bucket.name}/${encodeURIComponent(`someOtherPath/${fileName}`)}`;