How to pick a rechargeable battery?
This has the makings of an excellent question for tutorial purposes if you work on providing the needed extra information. 

A good starting spec, but really still too broad.
I'm answering this as below as the very fact that the answer is incomplete is (or should be) educational as to what you may look for and why in a battery. This list is incomplete but gives enough to start with.

Your Amp.hours add to about 3.3 Ah and you say about 10 Wh so your nominal mean system voltage is ~~ 3 Volts. Is that what you intended?

What application (apart from power curve). [Data telemetry / transponder / micro-cell site ...?}

What temperature range.
Unattended for how long? (day week year ...)
1 off or many?
Cost important?
General environment and temperature range?
Vibration? Safety? Mission Critical?    

How many days 'standby' needed.
Acceptable degradation rate with age.
System voltage?  

Consider the possible importance of various parameters in these purposefully OTT applications intended to stress various things:

My Lunar Rover needs a ....
My offroad racer ...
My polar explorer ...
My baby apnea alarm ...
My Boeing Dreamliner ...
My cardiac pacemaker ...
My elephant gastrointestinal whole of life remote monitor ...
My whale-mounted ...
My upper atmosphere balloon designed to drift for 10+ years ...
My deep-floating drifting marine telemetry buoy ...
My marathon runner data telemetry ...
My ultra low electrical noise instrumentation amplifier ...
My ...  

Think about the things that may matter is each case.
Note how even the mighty Boeing managed to get it wrong.   



Using LiIon and supplying current with a linear regulator you need 1 cellx ~= 3.3 Ah/day.
So a 3300 mAh battery would last about 1 day when new and not when not. It would provide 200-400 charge cycles. Double capacity to 6600 mAh and it would last up to 2 days but cycle life would rise. Use a 33 Ah cell and ... . Temperatures much under 0C would spil your day.  Max discharge rate of 281 mA is C/(3300/281) = C/11 so it would not mind.
If the winters were wintry and you used solar then you may only get 1 hour of charge/day. Charge rates of 1C would be OK but if your panel had to burst charge at say 5C when sun was available an average LiIon battery would be unhappy (at least). 

LiFePO4 would far better on most counts above but be beiger and weigh more. Which may or may not matter.

Lead Acid would not like peak charge rates and peak discharge rates would need Ah capacity derating somewhat from data sheet nominal. Self discharge and life time may be poor deep-ending on variant.

NimH ...

NiFe !!!! :-) ...

ie tell us more to help us tell you more.