A couple of issues...

if you use a LT 4054 to charge , then when your battery in low power that you cannot use a USB to power the DSO , because the LT 4054 check the battery is in low power , that it will be in small current mode to charge the battery to a hinger voltage then give out constant current to charge it , when you in low power , the DSO need a large current but the LT 4054 check the battery need charge in small current mode that it will consider the battery is bad and not work ,and then you cannot use your DSO … only if your turn it off and charge the it for a moment .

and if without a battery it will not work .