Removing a non linear noise trend from data
The trend can be perceived as some low frequency signal, so a simple highpass filter should remove it. It will naturally slightly decrease the peaks, but by very little, because such narrow peaks are mainly composed of high frequencies.

Here is some concept code in Python (using the NumPy, SciPy and Matplotlib libraries)

import numpy as np
import scipy.signal as scisig
import matplotlib.pyplot as plt
# assumes it has just two columns with f, r values
f, data = np.loadtxt('CQDataAbs.dat', unpack = True) 
# order4 IIR Butterworth highpass filter coefficients,
# the critical frequency is relative to the Nyquist frequency,
# determining the right value might require some experimentation
# note that using a higher order with such low cuttof will likely
# be numerically unstable, if you need higher orders,
# split it into several lower-order stages applied sequentially
b, a = scisig.butter(4, 0.0003, btype='highpass')  
# forward-backwards filter application with edge padding (mitigates edge defects),
# results in zero phase lag and an effective filter order 8 as it is applied twice
filtered = scisig.filtfilt(b, a, data)  
plt.plot(f, filtered, 'b.')
plt.show()


And here is the result


The shoot-up at the end may indicate that there are some higher frequencies in that area. Perhaps you can just disregard that region during fitting if possible.

If you were interested in the trend itself, you would obtain it by using a lowpass filter instead of a highpass filter. With zero phase lag application the result of highpass filtering should be almost identical (within numerical accuracy) to subtracting this lowpass trend from the original data.

Note that you may find that a different filter type might give you better results. A Butterworth IIR filter (with forward-backwards application to remove phase lag) is usually a good choice when the shape of the signal is important, because the frequency response within the pass-band is maximally flat. You could of course use some large FIR filter for better performance if you have enough processing power. With FIR filters you also don't have to worry about numerical instability that can occur in IIR filters of high orders.

Just in case you think about using a moving average (which is a FIR lowpass filter): That will likely perform terribly in this case, because in each window containing a peak the average will shoot up and subtracting that from the original data will greatly decrease the peaks. From a frequency perspective, the moving average filter does not mitigate higher frequencies very well, which are very significant within the region of the peaks.