Look at two BR2032 coin cells (or more in a series-parallel scheme). They will start at 5.4V and will remain above 5V for nearly the entire life of the batteries. They have a very flat discharge curve.
40mA sounds awfully high. In most jurisdictions, maximum transmitting power can be achieved at much lower current. And, most jurisdictions restrict the total amount of transmission time per hour (depending on application). An efficient antenna design plus a wideband preamp on the receiving end can greatly enhance range.
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