In pulsar timing you can't measure the absolute travel time from the pulsar to Earth. Instead you measure changes in the time of arrival relative to the expectation based on a model. You can think of it as changes in the time between pulses.
The Einstein delay covers changes in time of arrival due to the gravitational field of the pulsar and time dilation from its motion. Because the pulsar is moving relative to the Earth, its intrinsic rotational frequency is changed due to time dilation. In pulsar timing people will use Doppler shift to refer to a shift in the rotational frequency. Since we get one pulse per rotation the frequency of pulse arrival is Doppler shifted. (This is different than the frequency of the radio waves shifting, which also occurs.)
Assuming the pulses originate at the same location in the pulsar's atmosphere, then they all experience the same gravitational delay. If the pulsar is in a circular orbit, then it moves at a constant speed with a constant acceleration. This means all of the pulses will have the same motional time delay.
Since all pulses are delayed by the same amount, it is impossible to measure a relative change. That's why the Einstein delay part of $\Delta t$ depends on eccentricity, $e$. In an eccentric orbit the speed and acceleration of the pulsar change. Near periastron the pulsar moves faster and the time dilation is larger.
The post-Keplerian parameter $\gamma$ is the amplitude of the Einstein delay.
$$\gamma = e\, \left( \frac{P_b}{2\pi}\right)^{1/3}\,\left(\frac{G}{c^3}\right)^{2/3}\,\frac{m_c (m_p + 2m_c)}{(m_p+m_c)^{4/3}}$$
That's the maximum Einstein delay experienced during an orbit. The actual delay oscillates as a function of orbital phase. By fitting the amplitude of multiple post-Keplerian effects the mass of the pulsar $m_p$ and its binary companion $m_c$.