You shouldn't have to go landside and collect your bag. AA and Lufthansa have an interline baggage agreement, so what you should do is give the AA person at check-in all the information regarding your connecting flight, and they can tag your bags all the way through.

In that case you just follow the "flight connections" signs and go to T1 inside security ("airside.") When you get to T1 there will be airline counters where you can get your Lufthansa boarding pass.

But even if you had to go through UK immigration 4 1/2 hours surely ought to be ample, even with the current idiotic situation with the UKBA inspectors. I wouldn't expect to be an enjoyable 4 1/2 hours, however, especially the first 2 or so.

In addition to what Gardyloo said and definetly suggesting the same:
You can print your boarding pass for the connecting LH flight 23hrs before departure (London time) before you go to the airport in the US. That would 12.15pm minus 5hrs if you live in Eastern time zone, so after 7.15am on the day you depart from the US.

Confused about "connecting flight." The DFW-LHR flight is on AA booked with frequest flyer miles. I then purchased from a discounter the flight on Lufthansa from LHR to MUC. I didn't know that AA would check my bag through since I have two separate tickets. ?? At any rate, you think I have time to go through all the hoops and make the Lufthansa flight?

Where and how you purchased the tickets is irrelevant.
To make it simple, all major signature airlines (so this typically excludes low-cost carriers) have an agreement that they check the baggage thru to the final destination.

It is usually wiser to have all legs of one journey on "one ticket", though, as this is an extra safety net for tight connections (much less than your 4'15 hrs!). If you missed the connecting flight, you'd get transfered on later flights without costs.

But you have 4 hours and 15 minutes to get from Terminal 3 to Terminal 1.
If your plane from the US arrives on time or with up to 1.5 hours delay, and you have your baggage checked through, you will have nothing to worry.

As Gardyloo said, once you got off the plane from the US at Terminal 3, you follow signs for Flight Connections.

Here is a good step-by-step guide:
http://www.heathrowairport.com/heathrow-airport-guide/flight-connections/terminal-3-international-connecting-to-terminal-1-international

If you go one level up on the left hand navigation, you get more info on how to change flights at Heathrow.

There are big problems at the moment with the UK Border Control.
Lets hope it gets resolved soon.
http://bit.ly/immigration_heathrow

You REALLY REALLY want to ensure your luggage is checked through - even to the extent of contacting AA to try and add the Lufthansa leg to your existing ticket.

This means you are guaranteed to avoid the unpleasantness of UK passport control as if you leave them separate the AA check-in clerk may refuse to check your luggage through