We all join frequent flyer programs hoping to get "something for nothing," so this isn't the answer you want to hear, but if you like to fly first class, you'll either need to fly a lot more, make friends with someone who will gift you status or SWUs, or pay for first class tickets.
Frequent flyer travel, whether award seats or upgrades, is subject to availability. If an airline thinks it can fill a seat with a paying customer, it will always sell that seat rather than allow a frequent flyer to get it with miles. So even though the airline may sell, say, 2000 seats in a particular market daily, it may make only 100 of those seats available for redemption at the "Saver" rate. On top of this, the airline will want to reward people who give it regular revenue— their frequent flyers. When it comes to waiting lists, they will prioritize their elite flyers over passengers whom they don't "know."
When the agent told you that you had to fly on April 25, he or she was probably explaining that April 25 was the closest date available where any seats were still available to you; they are sold out for miles from the 26th through the 1st in first class. It is possible that AA will open additional seats to AAdvantage between now and departure, but there is no way of knowing.
AAdvantage does allow you to pay double the miles to get around the capacity control ("AAnytime" awards). But that still doesn't help you if the flights are sold out, and with the capacity cutbacks of the last several years (and more to come with the US Airways merger), many more flights are sold out today than in years past.
So your options are the same as those faced by anyone trying to redeem miles: make your dates more flexible, so you fly when those seats are available, or pay up.
Note that Business Extra is a separate program from AAdvantage. Business Extra points are not AAdvantage miles; you cannot transfer them to your AAdvantage account, and you cannot mix awards from the two programs (e.g. redeem a seat with one and upgrade it with the other). Unlike AAdvantage miles, you cannot purchase points to top off your account.
Per the Business Extra awards chart, you do not have enough points to redeem for a domestic ticket in first class, only for one of the capacity-controlled economy seats. Neither do you have enough points to award yourself Gold status (though if you won't actually fly 50,000 miles on American/OneWorld this year, that benefit would be largely wasted anyway). If you do book an economy seat, you will not be able to upgrade to first unless such an upgrade is offered for cash when you check in; it's happened to me on United before on an award ticket, but I don't know about American— their upgrade rules disallow upgrades on award seats.
It is only 650 points to upgrade from economy, but the deeply discounted fare classes (N, O, Q, S) are excluded, and all upgrades are subject to capacity controls (there must be A fare inventory available). I do not know enough about the program to know where a Business Extra upgrade stands in the upgrade pecking order.
With only 4019 miles in your AAdvantage account and no elite status, you have faint hope of redeeming for either an award seat or an award upgrade. Per the upgrade award chart, a one-class, one-way upgrade within North America is 15,000 miles, so with the current promotion that gives you 1000 extra miles for buying 5,000-14,000 miles, you'd need to buy 10,000 miles to upgrade one-way at 2.75¢ per mile plus 7.5% excise tax, which I make out to $295.63. That is just to request an upgrade, which may not be available currently, and for which you will be lower on waitlist than a frequent flyer or full fare passenger also attempting to upgrade with miles. It shouldn't take a frequent flyer to tell you that buying that many miles at that price for a potential domestic upgrade is an exceptionally bad value.
Per the reward seat redemption chart, it is 50,000 miles for a trip in domestic round trip, so you would need to top off with 42,000 miles (with the current promotion) costing over $2000, for something which may never become available for redemption— certainly not "worth it" for domestic first class.
As you have found, the 500-mile upgrade certs are only good on the very expensive Y or B fares unless you are AAdvantage Gold or Platinum (they are irrelevant for EXPs), and you are not. If you aren't going to reach Gold by your departure, these are irrelevant.