I'd like to stick with transistors ("bee-jay-teez"), that's supposedly where you weren't getting support.
Beta is h_fe, Yes.
I'm still assuming that the "common emitter" circuit is the basis of this discussion.
As for "dissipation by the resistors or transistors in the circuit", remember: current is the same everywhere in a series circuit. Let's do one thing at a time and not get caught up in a manic death-spiral.
If you want the transistor to do 240mA, then let's figure 300mA, plan for more than is needed.
I think that a beta of 50 is a good assumption.
300 mA / 50 = 6 mA
So, with "5V" as input voltage to our humble transistor, (5V - Vbe )/6mA = 4V / 6mA = 667?.
So, we'll live it up and go with 470?, working out to about 8 mA.
If V_cc is 9V, and I'll venture that V_ce might approach 300mV, then 9V - 0.3 = 8.7V.
8.7V / 240mA = 36?.
3 100? in parallel gets you 33?, resulting 263mA.
So, set that up, starting with > 2K for that collector resistor. Measure the voltage across that resistor. The voltage across the collector / the collector resistor ? = collector current. Replace the collector resistor with a lower value, paralleling for value as necessary, taking notes along the way, till you have several data points that you can analyse. [2K, 1K, 500?, 100?,...]
If you can manage with the milliammeter then go that route. Voltmeters make you get out the calculator, but they don't blow fuses.
The absence of an LED in the circuit doesn't matter, as we're just looking at, discussing, the collector current thing.