How to choose an off the shelf transformer for DC-AC inverter?
There are more degrees of freedom for the design of a ferrite transformer than for an iron one, so just having the data isn't necessarily sufficient, it's knowing what to do with it, and making some good assumptions.

First, get the data. From the digikey link, here is the core data and from google 'N27 ferrite pdf' here is the material data.

I've never designed a ferrite transformer before, as you'll see from the assumptions I make. But I'll point them out, and you can adjust for personal preference, and maybe I can receive education from more experienced board members. I know for instance that Andy's speciality is ferrite rather than iron.

You start with what flux to run at. With iron transformers, that's easy, 90% of saturation, as saturation is pretty much the only limit. With ferrites, heating is the limit as the frequency goes up. So we start with a guess, how much heat can we lose in the core? With a 150 Watt transformer, let's say 5 watts ASSUMPTION. Sounds reasonable efficiency and maybe low enough for a core of that size without doing any thermal modelling.

To get from dissipation to core flux, we need the operating frequency, the core volume, and the core loss graph. You suggested 200kHz. That's higher than N27 is recommended for, they suggest 25k to 150k, so I will do the sums for 100kHz ASSUMPTION as I don't like running at the extreme end of a range. The core data gives the core volume Ve as 17800mm3. The material data gives a graph on bottom left of page 5 (why don't they number the graphs?) of core loss versus frequency. 5 watts in 17800mm3 is 280kw/m3, which reading across to the 100kHz ordinate gives a core Bfield of a smidge over 100mT, let's use a round 100mT. Note there are two lines for different temperatures. Dissipation falls as temperature increases, which is good. This is a conservative number, the core is not going to run away thermally.

To get from core field and frequency to volts per turn, we need the core area. That's helpfully given in the core data as Ae at 173mm2. As saturation is not an issue, we don't need Amin, Ae the effective area is more appropriate. 

At 100kHz, the cycle time is 10uS, so a half cycle is 5uS, and the field slews -100m to +100m in that time, so it's 200mT/5uS, or 40kT/s. Multiply by the core area to get flux, 40k * 173u = 6.9 volts per turn.

With a 12v input, at 6v/turn, we need 2 turns for the primary (for H bridge drive) or 2x2 for push-pull.

Wire area? Let's assume you'll use the specified bobbin. They helpfully give the winding area on page 4 as \$A_N\$ = 210mm2. After winding inefficiencies (insulation, mis-packing, wires size not being exact) let's guess at 100mm2 left for copper. We allocate half each to primary and secondary. Two turns in 50mm2 means 25mm2 wire, or 12mm2 for push-pull.

With your 150 watt 12v input, that's about 12A. This gives a current density of 0.5A/mm2. I normally aim at 2-3 A/mm2 (in iron transformers, depending on all sorts of things) so this is so low I'm not going to compute resistive heating, it will be small. Had you noticed in pulling stuff apart how ferrite transformers would often have very under-filled copper windows.

You could use this very low current density to say that 5W core loss is too much, aim for a lower field and lower volts/turn, with more turns. That's for a later iteration. In broad brush terms, the design looks plausible. Now for a few of the confounding details.

Significant differences between iron/mains and ferrite transformers include skin depth and proximity effect. Although the current density is apparently very low, the skin depth effect will increase the effective resistance and create more heat than the DC value suggests. The 25mm2 winding is not just impractical to wind and terminate if it stays one fat wire, it's actually better in skin effect terms to make it from multiple parallel strands of thin wire. While you can do the sums on skin depth INCOMPLETE, I have no experience of the sort of factors that should be used for proximity effect at 100kHz, but believe them to be relatively small, less than a factor of 2 ASSUMPTION, perhaps those more experienced could enlighten me.

Gapped versus ungapped? We haven't computed the magnetising current. With a gapped core, we have a well-defined reluctance, and so can do a good job. With an ungapped core, the magnetising current can range over a wider range, as the permeability of the cores varies and the microscopic residual air-gap varies, but even with its wide range, the mag current is lower than that with a gap (compare \$A_L\$ (inductance of a single turn) 3300nH+30%-20% ungapped plays 862nH gapped). So do you want an accurately known big current, or a roughly known small current? 

Let's compute it for the gapped core anyway, to see what ball-park we're in. The data sheet gives the gapped core an effective permeability, \$\mu_e\$ of 407, and a Le of 103mm. As B=uH, H=B/u. The H field needed for 100mT with that \$\mu_e\$ is \$\frac{0.1}{4 \pi 10^{-7} 407}=195\$ A/m, which over a length of 103mm and for 2 turns is a current of 10A.

Compared with your load current of 12A, a 10A mag current would be quite a lot with a gapped core. With an ungapped core, you could expect it to be in the region of 862/3300 times that, or around 2.6A(-30%+20%), somewhat more reasonable. 

That's as far as I'm going to go. The core size you've chosen looks a little overkill based on the apparent current density, but not by much, and when you need to derate for RF losses, perhaps not at all. You may want to drop the peak field to reduce the magnetising current and/or core dissipation. Given my assumption on power dissipation, and no design on RF losses, you would need to build and test something.

I've demonstrated the minimum data you need to pull out of the data sheets, and how to use them. I've pointed out where my experience is lacking, and where you will have to dig further, iterate further, and experiment.