TL;DR — throttleable indeed, quite deep throttling in fact

Thrust at Max / Nominal ⟶ 3750lb
Thrust at Throttled Minimum ⟶ 330lb

Nb numbers for the Minimum Thrust are a little squirrelly having also found mentions of 220lb thru 370lb, note ANY of those’d be quite impressive vis à vis deep throttling, but 330lb seems most common in more knowledgeable analyses, notes in the second link have more in that point

NASM ⟶ 330lb Min RE 509A
WalterWerke.co.uk ⟶ 330lb ALL Single Chamber 509x

Prof BAXTER has quite clearly had hands on with original hardware, torn it down, and analysed it quite closely, likely also gone over Luftwaffe docs etc, and c1947 that’d make sense. Also note BAXTER is focussed in the 509A2 variant. Further, the primary source referenced for the above was the first of the two below, and from which the diagram was YOINK’d.

Note throttle upper RHS, that controls the boxes upper RHS and centre which in the text are called the Turbine Speed Control Valve and Main Regulating Valve respectively, both of those are continuously variable ie pilot can smoothly increase and decrease thrust to anywhere between the MIN and MAX thrust numbers noted earlier.

F11 DIAGRAM of OPERATING SYSTEM for 509 A.2 MOTOR

AIRCRAFT ROCKET MOTORS with Special Reference to German Developments with Hydrogen Peroxide (10.1108/eb031538)
A D BAXTER ⸱ Aircraft Engineering ⸱ No 19 ⸱ Vol 8 ⸱ c1947

PROSPECTS and PROBLEMS of Rocket Propulsion for Aircraft
Aeronautical Journal ⸱ Vol 59 ⸱ No 533 ⸱ c1955 Prodessor A D BAXTER ⸱ 10.1017/S0368393100117560