In a nutshell: Yes, you can keep your water with a thick atmosphere but you've set yourself a difficult scenario with a lot of hoops to jump through to survive.
There is a lot in your question, I'm going to take some of the bits separately to discuss them:
Rocky planet with similar mass to that of Earth's, orbiting a red dwarf star. It is tidally locked with no natural satellites.
Given the tidal locking, similar earth mass and orbiting a red dwarf things are already looking risky, you need a strong magnetic field which in turn requires rotation for a magnetic dynamo. The faster the rotation, the stronger your field. We're rotating rather slowly in our several days of orbit.
However if you increased the size of the liquid iron core (and probably the size of the planet a little) then we could get a largish magnetic field due to the convective flow of material within the planet.
But now we come to the red dwarf, you've got very little chance of your atmosphere surviving the coronal mass ejections. The red dwarf calms down later in its life though, so perhaps - if we assume your atmosphere survived - we can discuss other methods.
For that to be possible, I explored the possibility of a denser atmosphere, perhaps a higher content of CO2, ocean and winds that diminish the temperature differences between night and day sides, tidal heating, volcanic activity, and strong mantle convection.
Yup, a stronger global warming effect would definitely help (though you need to have kept a thick atmosphere for this). Your strong volcanic activity could help with the production of CO$_{2}$ and other gasses.
Since it orbits a red dwarf star, a strong magnetic field must exist to prevent the planet's atmosphere from being stripped away by solar winds. Though it wouldn't be a flare star, as a red dwarf, I think it would still be less stable in terms of luminosity than our sun. Despite having slow rotation, could the planet's powerful magnetic field be justified by strong mantle convection and plate tectonics?
As I mentioned above, we can (potentially) hold onto the atmosphere if we have a large liquid iron core and strong convection. You should aim to hold onto the atmosphere until your star calms down and then use volcanic activity to repopulate your atmosphere.
It has a year of roughly 15-30 days, and though it doesn't have any natural satellites, I'm considering another rocky planet with an orbit close enough to exert a gravitational pull that would cause strong tides (hopefully making a liquid ocean on both sides of the planet more feasible).
This wouldn't be a stable system, the two planets would eventually fall in towards one another. In this case, too, your planet would in fact be a dwarf planet since it hasn't sufficiently cleared the surrounding area.
I'm also considering an axial tilt that would allow inhabitants to measure time through seasons in the year-long day.
You would still get seasons since the rotational axis of the planet is stationary relative to the orbital axis.
I'm still very much an amateur at this.
None of us are "professional", searching the internet for relevant information is a skill but everyone has it to some degree. It just takes practice.
Summary
You need:
- Large liquid iron core to provide the convection required to support a magnetic field and warm your planet a little.
- Your atmosphere to survive the early years until the star has calmed a little, you can also use your strong volcanic activity to repopulate the atmosphere.
- A thick atmosphere this provides the global warming effect and the gasses will mix, providing high winds which keep the temperature comparable on both sides.
- To be slightly further out in the habitable zone since your atmosphere will keep you warm, you don't want to be too warm and evaporate off your water or atmosphere (though a slightly higher mass will help here).