You may have two problems.
First, the SFP is transmitting out of spec. At most, it should put out -1.2dBm per the link supplied by lacasitos above. You may be able to RMA it if a Cisco SFP is complaining that another Cisco SFP is sending too hot of a signal.
Second, because you're now getting a Rx threshold error (it should not put out less than -7.3dBm and -13.8 is quite a loss on a short link), the fiber may not be fully inserted at either the tx or rx end, or the fiber is dirty or the transmitter or receiver is dirty. (It's usually the fiber getting scratched on insertion unless you're in a dirty environment.)
If RMAing is out of the question, you could always introduce some loss to bring it under spec by using an attenuator or introducing extra patches.
Comments converted into answer: Please spend a bit of time learning the actual terminology rather than inventing your own, which confuses both you and us. An SFP (small form pluggable) is not a connector. It typically HAS one or more often two female optical connectors as part of itself, but what it IS is an electro-optical transceiver. The 10 Gig ones are not SFPs, they are SFP+ The optical cable does not matter if it's the right family and clean, what you need to is know what the SFP (or SFP+) that you are connecting to at the switch is, and get one compatible with it. Clean, by the way, is a VERY big deal with single mode fiber connections.
And the SFP+ you have shown (without showing the relevant part where the cables plug in) most likely needs a pair of optical cables, not a single cable. If it has two optical connectors, one is transmit, one is receive. Transmit at one end connects to receive on the other. I'd guess that 10GE-LR is intended to be 10GBASE-LR, which is 1310 nm lasers on single-mode fiber up to 10 kilometers, not a bidirectional standard.
https://en.wikipedia.org/wiki/10_Gigabit_Ethernet
The cable test tag on your patch cable showing what its insertion and return losses at various wavelengths were when it shipped from the factory is utterly irrlevant to the choice of the correct SFP+ - which seems to be your question, buried under a lot of mis-used terminology you need to learn if you have anything to do with this equipment.
Best Answer
Short answer: NO
The bidirectional (single wire) SFPs come in pairs because they use different frequencies in each direction. Even if you did split the fiber, the TX and RX wavelengths would (a) be different, and (b) not likely to conform to any standard SR/LR optics. You'll need some form of "media converter" (aka: two port switch) that can take the matching SFP and convert it to whatever standard you want.