Converting GeoTIFF with GCPs into PROJCS with python/GDAL?

I have some data with the following projected coordinate system (obtained via gdalinfo)

PROJCS["WGS 84 / UTM zone 11N",
GEOGCS["WGS 84",
    DATUM["WGS_1984",
        SPHEROID["WGS 84",6378137,298.257223563,
            AUTHORITY["EPSG","7030"]],
        AUTHORITY["EPSG","6326"]],
    PRIMEM["Greenwich",0,
        AUTHORITY["EPSG","8901"]],

    UNIT["degree",0.0174532925199433,
        AUTHORITY["EPSG","9122"]],
    AUTHORITY["EPSG","4326"]],
PROJECTION["Transverse_Mercator"],
PARAMETER["latitude_of_origin",0],
PARAMETER["central_meridian",-117],
PARAMETER["scale_factor",0.9996],
PARAMETER["false_easting",500000],
PARAMETER["false_northing",0],
UNIT["metre",1,

    AUTHORITY["EPSG","9001"]],
AXIS["Easting",EAST],
AXIS["Northing",NORTH],
AUTHORITY["EPSG","32611"]]
_EPSGProjection(32611)

I have other files from the same source but these have headings like:

Coordinate System is `'
GCP Projection = 
GEOGCS["WGS 84",

DATUM["WGS_1984",
    SPHEROID["WGS 84",6378137,298.257223563,
        AUTHORITY["EPSG","7030"]],
    AUTHORITY["EPSG","6326"]],
PRIMEM["Greenwich",0],
UNIT["degree",0.0174532925199433],
AUTHORITY["EPSG","4326"]]
GCP[  0]: Id=1, Info=
      (0.5,0.5) -> (-118.624137731569,34.3543955968602,0)
GCP[  1]: Id=2, Info=

      (1481.5,0.5) -> (-118.503860419814,34.3500134582559,0)
GCP[  2]: Id=3, Info=
      (1481.5,1016.5) -> (-118.504350292702,34.2792612902907,0)
GCP[  3]: Id=4, Info=
      (0.5,1016.5) -> (-118.624724103593,34.2838459399194,0)

How do I convert the second example into something like the first example with a PROJCS entry using Python/GDAL.

If they can't easily be converted, what would the equivalent of this code that sets up the coordinate system in python be for the second example?

ds = gdal.Open(fname)
data = ds.ReadAsArray()

gt = ds.GetGeoTransform()
proj = ds.GetProjection()
inproj = osr.SpatialReference()
inproj.ImportFromWkt(proj)
projcs = inproj.GetAuthorityCode('PROJCS')
projection = ccrs.epsg(projcs)

Update 1:

gdal version is 2.1.12

gdalwarp -t_srs epsg:32611 file2.TIF test.tif

then gives the following:

Coordinate System is:
GEOGCS["WGS 84",
DATUM["WGS_1984",
    SPHEROID["WGS 84",6378137,298.257223563,
        AUTHORITY["EPSG","7030"]],
    AUTHORITY["EPSG","6326"]],
PRIMEM["Greenwich",0],
UNIT["degree",0.0174532925199433],

AUTHORITY["EPSG","4326"]]
Origin = (-118.624740866712244,34.354482507403588)

That doesn't really help though as 1) It hasn't added in the PROJCS entry 2) I don't necessarily know the epsg value to enter for all of the files I have.

Update 2:

Doing this:

gdalwarp -s_srs epsg:4326 -t_srs epsg:32611 file2.tif test.tif

Works and means that the new file now contains the PROJCS information in the header.

PROJCS["WGS 84 / UTM zone 11N",

However the issue is that I only know this file needed 32611 as I had an example of another image from the same area. How would I add the projection information when I don't know what it should be beforehand?

Update 3:

I've used the utm library to find the utm zone and using that has allowed me to use gdalwarp. The bit of info I was missing is that the EPSG code is simply 326/327 + utmzone.

gdalwarp -s_srs epsg:4326 -t_srs epsg:32604 file1.tif output.tif

However, this fails for some files

ERROR 1: Too many points (441 out of 441) failed to transform,
unable to compute output bounds.

This data is from Alaska so fairly high latitude but not sure why it'd fail?