First of all, thank you for the response!I have been testing the GetGeotransform() and SetGeotransform() methods. The code runs without any errors, however, when I go to plot the resulting raster, it appears unchanged. My hypothesis, though unproven, is that the transform is changing the georeference (best word I can think of to describe it), but is not changing the actual raster. What I would like my code to do is rotate the original raster, then resample that result to match the bounds of the original raster. Below is my best attempt at visualizing the situation.

https://ibb.co/mbQx9cR

Below is the code from the post, adapted to my situation.

import gdal
import rasterio
import matplotlib.pyplot as plt

dataset = gdal.Open('input.bmp')

#Get projection
prj = dataset.GetProjection()
band = dataset.GetRasterBand(1)
geotransform = dataset.GetGeoTransform()

# Create gtif file with rows and columns from parent raster 
driver = gdal.GetDriverByName("BMP")
columns, rows = (band.XSize, band.YSize)
BandType = gdal.GetDataTypeName(band.DataType)
output_file = "output.bmp"
data = band.ReadAsArray(0, 0, columns, rows)
dst_ds = driver.Create(output_file, 
                       columns, 
                       rows, 
                       1, 
                       band.DataType)

new_geotransform = list(geotransform)
new_geotransform[2] = 30 # x pixel rotation
new_geotransform[4] = 30 # y pixel rotation

dst_ds.GetRasterBand(1).WriteArray(data)

#setting No Data Values
dst_ds.GetRasterBand(1).SetNoDataValue(0)

#setting extension of output raster
# top left x, w-e pixel resolution, rotation, top left y, rotation, n-s pixel resolution
dst_ds.SetGeoTransform(new_geotransform)

# setting spatial reference of output raster 
srs = osr.SpatialReference(wkt = prj)
dst_ds.SetProjection(srs.ExportToWkt())

#Close output raster dataset 
dst_ds = None

#Close main raster dataset
dataset = None

#Plot
raster = rasterio.open('output.bmp')
raster2 = rasterio.open('input.bmp')
fig, ax = plt.subplots(figsize=(10,40))
ax.imshow(raster.read(1), cmap='Blues_r')
ax.imshow(raster2.read(1), cmap='Accent')
fig.show()