Finding delay where squared error between signal and data is minimal
Just doing something like:
$$
e_\delta = \sum_{n=0}^{N-1} \left| x[n-\delta] - x_\delta[n]\right|^2
$$
for several values of $\delta$ and choosing
$$
\hat{\delta} = \arg \min_{\delta} e_\delta
$$
(i.e. the value of delta that minimizes $e_\delta$) will probably be good enough.
Example signals below:

gives the following error plot

The error plot seems to give the right value of about 5.5 samples for this contrived example data.

Code Only Below
from numpy import log10, asarray, polyfit, ceil, arange, exp, sin, pi, log, random, sum, diff
import matplotlib.pyplot as plt
from scipy.signal import kaiserord, lfilter, firwin, freqz

T = 100
time_period = list(arange(1,T))

K1 = 10
K2 = -10
tau = 200
measurement = [K1 + K2*exp(-x/tau) for x in time_period]

fir_filter = [0,0,0,0,0,0.5,0.5,0,0,0,0]
channel_1 = lfilter(fir_filter,1, measurement)

max_delay = 20

def error_calculation(signal, delayed_signal, delay):
    error = [ (signal[time-delay]-delayed_signal[time])**2 for time in list(arange(delay,T-delay)) ]
    return sum(error)

plt.figure(0)
plt.figure(figsize=(20,20))
plt.plot(time_period, measurement)
plt.plot(time_period, channel_1)
error = []
for delay in arange(1,max_delay):
    error.append(error_calculation(measurement,channel_1,delay))
    
plt.figure(1)
plt.figure(figsize=(20,20))    
plt.plot(arange(1,max_delay), error)