Can I invoice for future start dates?

First off, there are accounting rules, then system limitations and finally company/departmental policies.

The accounting rules say that you may be able to do it. Unfortunately your accounting system may not have the ability to do it. Even if your accounting system says that you can do it, You may have an analyst or accountant where the policy is to look at what was billed and then setup a schedule to recognize 1/12 each month as revenue.

Technically from an accounting point of view, you can bill well in advance of 1/1/2020 and it would be a prepaid or a deposit receivable and that is held within AR. Once its is paid (in advance of 1/1/2020) it would become a deposit or a prepaid liability. Then every month you would recognize 1/12 (assuming its a 12 month contract and that its based on time and not other factors like usage etc.)

You can certainly send out an invoice prior to the start date. Consider the example that for a loan company, they may give the borrower a coupon payment book to be used to send in a coupon with each future payment. Those are invoices. The key here, as you point out, is that you cannot record the accrual of income until it is actually due, which is 2020. Consider that where payment is due the first of the month (such as in a lease) the invoice MUST be sent out before the first or else it may arrive to late for the lessee to make payment by the first of the month.