The good news is that you'll have plenty of daylight for your ambitious evening. But if this is the day you actually arrive from the US, the bad news is that you might be too tired to appreciate being up that late.
If jet lag isn't a problem, I'd suggest saving Tower Bridge until the evening. When you've seen the Tower in the morning, work your way west to St. Paul's through the City and stop for lunch somewhere (I recall a pub called "Hung, Drawn and Quartered," which is certainly inspiring but I can't vouch for the food). You should have time for a look at the Roman Temple of Mithras (on Queen Victoria St.) and/or the Museum of London (a few blocks north of St. Paul's). Three hours should be more than enough for St. Paul's before evensong, keeping in mind that the service itself is part of your experiencing that magnificent place.
To "kill" your three evening hours, I'd recommend walking south from St. Paul's across the Millenium (pedestrian) bridge to the south bank, then east past the reconstructed Globe Theatre, the replica of Drake's ship "Golden Hinde," and other sights, down to Tower Bridge. Admire it and take pics in the evening light. Cross the bridge to return to the Tower in time for the Keys. Somewhere along the way stop for dinner, maybe just a quick bite on a bench overlooking the river.
That's a long busy day but an efficient way to see a lot if you have the energy!
