A voltage controlled switch for a single NiMH cell
The problem is that while charging, the MAX chip outputs up to 5V into
  the NiMH battery, which would be too high of an input voltage to the
  TPS61006.


The DS2711 uses an external PNP transistor to regulate charging current. In your circuit this transistor's Emitter is connected to +5 V USB, which is why the battery voltage could potentially go up to +5 V (if the battery became disconnected or went open circuit internally). So you just need a way to prevent the charging voltage from going over 3.6 V (maximum allowed input voltage of the TPS61006). 

The simplest solution is to power the current regulating transistor from a voltage regulator that puts out 3.6 V or less. A 3.3 V regulator would provide ~3.3-1.7 = 1.6 V of 'headroom' between the charging input and battery voltages. 

The circuit looks like this:-







simulate this circuit &ndash; Schematic created using CircuitLab

At 500 mA (a realistic maximum charging current for a AAA NiMH cell) the 3.3 V regulator would have to dissipate ~(5-3.3)*0.5 = 0.85 W. Without the voltage regulator Q1 would have to dissipate this power, so total dissipation is the same but it is now spread over two devices. The FCX718 has a maximum recommended power dissipation of 1 W on a 15 mm2 PCB area. If used alone it might have to dissipate up to ~(5-1.2)*0.5 = 1.9 W at 500 mA, whereas powered through the 3.3 V regulator it would only need to dissipate a maximum of ~(5-3.3)*0.5 = 1.05 W.

Another potential issue is the charging current sense resistor (R5 in my circuit), which drops up to 0.125 V. You could use a standard 2.5 V Gate drive FET to bypass this resistor in battery mode. The FET would become effective once the booster had started up, permitting higher battery current draw without extra voltage drop across the sense resistor.

The circuit for that could look like this:-







simulate this circuit

Q1 keeps the FET turned off in USB power mode. In battery mode the FET is operating in reverse, with Drain and Source functions swapped.