Here's a spacetime diagram that shows a ship (moving with $v=(3/5)c$ in the lab frame) sending periodic signals to the lab frame. Time runs upwards.
I've drawn the diagram using my "rotated graph paper" method so that it is easy to count ticks along segments. The light-clock diamonds are traced out by the light-signals in a light-clock along the timelike diagonal of the diamond. The important feature is that all light-clock diamonds have the same area.
You can see the velocity of the ship is $\displaystyle v=\frac{T_1E_1}{OT_1}=(3/5)c$ (count 5 ticks up , then 3 space-ticks ["sticks"] over). By construction (using these numerically-nice numbers), there are 4 diamonds along the ship worldline segment $OE_1$.
The time dilation factor is $\displaystyle\gamma=\frac{OT_1}{OE_1} =\frac{5}{4}$, which you can check using the formula $\displaystyle\gamma=\frac{1}{\sqrt{1-(v/c)^2}}$.
You can see the Doppler factor $\displaystyle k=\frac{OR_1}{OE_1}=2$ when they are separating,
which you can check using $\displaystyle k=\sqrt{\frac{1+(v/c)}{1-(v/c)}}$ with relative-velocity $v=(3/5)c$.
You can see the Doppler factor $\displaystyle k=\frac{OR_{-1}}{OE_{-1}}=\frac{1}{2}$ when they are approaching,
which you can check using $\displaystyle k=\sqrt{\frac{1+(v/c)}{1-(v/c)}}$ with relative-velocity $v=-(3/5)c$.
In other words, when the ship is emitting with a period of 4-ticks (in the ship frame),
the lab receives the signals (on the lab-worldline) with period 8-ticks when they are separating [thus, redshift],
but period 2-ticks when they are approaching [thus, blueshift].
In both the separating and approaching cases,
the lab regards (by, say, a radar measurement to determine the events on the lab-worldline are simultaneous with distant events on the ship worldline) the elapsed time between consecutive ship-emission-events to be 5 ticks.
So, by proportions, the answer to
Will the time interval of the laser pulse clock (with a 1 second time interval according to the space ship) received by the stationary observer ALWAYS be greater than 1 second due to time dilation, even if the spaceship is moving towards the stationary observer?
is no.
As mentioned in my comment to the OP,
consult Time dilation in an approaching object - contradictory to Doppler's effect?
for some details on "time-dilation" vs "Doppler effect"
with regard to spacelike-related events vs lightlike-related events.
If we turn-off time dilation (assuming absolute time),
we get a diagram like this:
For simplicity, we keep the emission signals the same.
So, now, assuming no time-dilation (pure classical Doppler effect),
the emission-period by the ship is now 5 ticks.
- When they are separating, $\displaystyle k_{classical,sep}=\frac{OR_1}{OE_1}=\frac{8}{5}$,
which is consistent with $(1+(v/c))$ with $v=(3/5)c$.
- When they are approaching, $\displaystyle k_{classical,app}=\frac{OR_{-1}}{OE_{-1}}=\frac{2}{5}$,
which is consistent with $(1+(v/c))$ with
$v=-(3/5)c$.
[NOTE: Without time-dilation, this experiment will not
be in agreement with the principle of relativity.
Consider light-signals emitted by the lab frame every 5 ticks.
When does the ship receive those signals?
]
UPDATE to address the OP's question in the comment.
Here's a radar measurement by the lab frame.
Wristwatches zeroed when the lab and ship meet at event O.
After 2 ticks on the lab-wristwatch (event $S_1$), the lab sends a signal (encoded with "2"), which meets the ship worldline (at $E_1$) and is reflected back to the lab, received when the lab-wristwatch reads 8 ticks.(at $R_1$).
Data collected by the lab:
$OS_1=2$, $OR_1=8$
Data analyzed by the lab:
$k=\frac{OE_1}{OS_1}$, $k=\frac{OR_1}{OE_1}$ (with $E_1$ possibly unknown).
So, $k^2=\frac{OR_1}{OS_1}=\frac{8}{2}=4$. Thus (Doppler Effect) $k=2$, predicting $OE_1=(k)OS_1=4$.
The lab assigns time-coordinate $OE_t=(OR_1+OS_1)/2=5$ to the event $E_1$.
The lab assigns space-coordinate $OE_x=(OR_1-OS_1)/2=3$ to the event $E_1$.
Thus (Time Dilation) $\displaystyle \gamma=\frac{OE_t}{OE_1}=\frac{5}{4}$.
(We could suppose that when the ship receives the light-signal, it sends its wristwatch reading "4" with the reflection [of the "2" from the lab].
Thus the reception event at the 8th-tick has the original "2-tick" timestamp carried in the light-signal from the lab and possibly the "k*2-tick" timestamp from the light-signal from the ship.)
As an exercise, you should carry out an analogous radar measurement done by the ship. Here is the diagram.
You should also use the very first diagram to analyze the situation when they are approaching, using a radar-measurement beginning at an event 8-ticks before they meet (when their wristwatches read $-8$).
(You are encouraged to construct analogous diagrams for $v=(4/5)c$.
[Velocities with rational Doppler factors lead to relatively-simple arithmetic with fractions.])
