If the incoming flight is on time, LH will (rightfully) deny any claim for comp. for delay if you miss the SFO flight because 45min at FRA is a valid and doable connection. As stated above if your incoming flight is on time and you head straight to the gate (presumably it will be an easy A to Z gate) there is a high chance that you will make the connection, but there is no time for lounge, window shopping, excessive toilet visit etc.
If you miss the SFO flight (due to incoming flight arriving FRA late) LH will rebook. Go to the transfer desk. If you prefer to travel the next day and LH agrees to this, it will not prevent you from getting EU 261/04 comp. for delay (unless the delay was caused by extraordinary circumstances but that is a diff. issue).
However, if LH is able to rebook you to SFO arriving with less than 4 hours delay and offers this to you, you are obviously not entitled to comp. if you decide for a different later flight (which arrives SFO with a delay of 4 hours or more) and any hotel, meals etc will be on your own dime. I find it rather unlikely that LH will be able to rebook you to SFO with a delay of less than 4 hours but may certainly be able to rebook you same day which will make your claim for reimbursement of a FRA hotel difficult (if you decide to be rebooked to the following day).