I have used two different online coordinate converters to convert Lambert coordinates 386.1/1185.0 to Lat/Lon and receive an answer 47.42715241/-5.62147696. This reference is 426 kms on a bearing of 242 deg from the known position of the place to which the Lambert reference is given.
I think you may have used the wrong dimensions. Projected Coordinate systems usually take meters as units, and your point should be in the North of France (Lambert Nord, Lambert 1 has an additional shift of 1000000 Northing).
If you enter coordinates in kilometers, they will end up at the origin of the projection, which is off the Atlantic Coast:
So try 386100 / 1185000 and you should get the right degree coordinates.
I believe there is a confusion between EPSG:27561 and EPSG:27571. Here are the differing parameters between the two:
EPSG:27561 NTF (Paris) / Lambert Nord France False Northing: 200000 m EPSG:27571 NTF (Paris) / Lambert zone I False Northing: 1200000 m
That would explain the differing latitude values. I think the online converters may be returning longitude values relative to the Greenwich prime meridian rather than Paris one (Thus NTF or WGS 1984 not NTF (Paris)).
I haven't check it, but there could be a grads versus degrees issue as well.
In the Esri projection engine, if I use 27571, the results are
54.808375149 -3.2706144771 (NTF (Paris), grads)
If I use 27561,
64.7429894875 -4.0008561734 (NTF (Paris), grads)
Ah, even with changing the values from grads to degrees, I'm still not matching your results, so without checking the online converters, I don't know if this completely answers your question.