mm per square meter is a measurement of volume, it's not really a depth divided by an area. You can tell as much because if you have 1 mm per square meter and you increase the coverage area to 2 square meters, the depth doesn't change---the volume increases. You just need to convert your volume measurement.
$$
1.2 mL = 1.2 cm^{3}
$$
1.2 cubic centimeters is equivalent to a 1.2 cm depth over an area of 1 square cm.
$$
1.2 cm = 12 mm
$$
$$
1 cm^{2} * \frac{1 m^{2}}{100^{2} cm^{2}} = 0.0001m^{2}
$$
12mm per 0.0001 square meters is not a useful metric. You have to obviously make the square meters 1 square meter, but you have to be careful. When you increase the area, the depth will necessarily decrease (because the volume has to be equivalent). Multiplying the area by 10,000 square meters per square meter will give you 1 square meter; subsequently, the depth must be divided by the same amount.
$$
\frac{12mm}{10,000} = 0.0012 mm
$$
So you have 0.0012 mm per square meter with each tip of the rain gauge. We can check our work by converting it back to a normal volume.
$$
0.0012 mm * 1 m^{2} * \frac{1000^{2} mm^{2}}{1 m^{2}} = 1200 mm^{3} = 1.2 mL
$$
As a side note, here is a useful tool provided by the United States Geological Survey for converting depth per area measurements to equivalent volumes:
https://water.usgs.gov/edu/activity-howmuchrain-metric.html
Edit
Going back to the depth given by the data sheet, I ran the numbers and
$$
(0.2794mm)/(1000mm/m)*(0.0055m^2) \approx 1.54 mL
$$
which about matches the volume you have. So if you want the depth for one square meter, you just have to multiply the area by the factor of $\frac{1m^{2}}{0.0055m^{2}}$ and divide the depth by the same factor. So essentially you want to multiply the depth by the catchment area of the gauge.
$$
(0.2794 mm) * (0.0055 m^2/m^2) = 0.00154 mm
$$
as the depth for 1 square meter.
