Hmm for the form where the client can add line items, I'd suggest checking out Fillout.com (https://www.fillout.com/). They integrate with Airtable and support line item functionality, which Airtable forms currently doesn't
The idea would be you'd have a table set up just to have these form submissions, and it'd have a linked field to Line Items. You could also have a linked field to the Clients table here but that's probably a security / privacy issue, so I'd recommend just getting your clients to key in their emails and using an automation to match it against the Client record. That form would thus allow them to identify themselves via email, and create multiple line items via one form submission
In the "Line Items" table, you would:
1. Create a formula field that will output the month year of the created time of that record, i.e. the submission date, and it would look like "Feb 24" or something like that.
2. Create a formula field to combine that with the client's email address (or some other unique identifier), to come up with "test@example.com - Feb 24"
3. Create a linked field to "Invoices"
4. Create an automation that will trigger every time a "Line Item" record is created, and its action will be to paste the value from the field from step 2 into the linked field to Invoices
This will create one invoice per client per month, with all of the line items they've submitted across the month