I can't just use an amp with the 3.5mm jack?
You can but it's not optimal. From my experience using earphone outputs* the levels are either too low then if I increase the volume it sounds either a bit "off" or there is distortion, like when the vocals going up an octave or two it sounds more grating than the same recording on a line output. The popular explanation for that is that the amp circuit has some distortion (which increases with the volume setting) and the external amp amplifies that as well; other things to consider are the specs on the earphone output (which is different from the impedance and voltage specs on a lineout, even with variable line outputs controlled only by a preamp but doesn't go through an amplification stage) and the input on the amp (usually designed for the specs on a line output).
If you really have to use the 3.5mm output you'll need an amp with a lot of power and high gain, that way you can keep the volume on the source lower and just use the amp to get the volume higher (as well as driving the headphones/IEMs) with a weaker but cleaner input signal. Most amps I've tried as such however had too little gain that I'm using them close to maximum setting and the volume is just about right, if the amp isn't already distorting at that point too.** That was all with an HD600, so I'm guessing your IEMs are not likely to encounter the same issue,
but again, do they need that kind of power as the Tomahawk or any portable amp for that matter? Note that I only recommended those other DAPs because I thought you were going to use an older click wheel iPod (which I thought was what $150 can buy, not knowing it's for a lower memory Touch for Spotify), and if the output power level is useful, it will not necessarily be a lot of improvement with those IEMs, but with other headphones or custom IEMs that you can try later on (and such players likely might not need any help from another amp).
Here's how I would go about all that if I was going to listen primarily through Spotify, and let's assume I've already got the 5g Touch,*** I'll just use that with the IEM on its own for the most part, especially when I'm on the go, and maybe even at my office desk. When I settle down into a fixed listening position (my desk at home; not so much at work since I have the XBA-4), I'll use that with the Apple dock that has the DAC chip in it, then use the dock's lineout to direct the signal to an amplifier like the Ibasso PB-1 or a similar RSA (with the input at the rear, since I'm used to home set-ups that have the cables all in the rear, save for the headphone's) and a headphone that will really benefit from the amount of power as well as the capacitance of such amplifiers.
*vs the actual lineout on the same device - Touch 3g, 4g; 4g, 5g wheel iPod; also devices that don't have a lineout, like one of my old HDD Creative Zen, smartphones, etc
**a CMOY, but this was the DIY-er's first version; his second version had higher gain I think, and also an active ground
***because honestly I'd rather get a smartphone or a larger tablet which I would have with me anyway; if I wanted a dedicated player smaller than a tablet, well, I'd get a Fiio or Ibasso