Can a non-orientable closed surface of odd genus be immersed in $\mathbb R^3$ so that the associated height function be of Morse-Bott type and have no centers?

That is, the height function would have only Bott-type (circles) extrema and saddle singularities.

My intuition is that no. (I've asked this question on math.SE but did not get any answer.)

Consider the projective plane $\mathbb RP^2$ as the Boy surface (left) and the Klein bottle $K^2$ (right):

(image from the book). The 8-shaped level sets are immersions (i.e., not self-intersecting) except for where singular points are shown. The vertical line in the right-side figure is a homologically non-trivial cycle.

For even genera $g$ (except $g=2$, which is a different story), it is easy to do: e.g., connect the top and bottom of $K^2$ (right) by a tube (as if you drill a wormhole along the vertical axis), which will form a surface of genus $g=4$ immersed with two Bott-type extrema (circles) and two Morse-type saddles. (You can get any even genus $g\ge4$ by adding more handles.)

However, adding such a handle to the $\mathbb RP^2$ (left) seems not possible. Suppose you add such a handle connecting the bottom to the top of the figure (left). There must be a singularity on that handle. Indeed, consider the evolution of the level sets from the bottom to the top along this handle. The level sets at its endpoints are circles $S^1$ immersed in the plane: O-shaped at the bottom and 8-shaped at the top, which are not regularly homotopic by the Whitney–Graustein theorem. Therefore, there must be a singularity in between.

My intuition is that the singularity will be similar to the saddle shown on the picture (roughly speaking, not preserving the parity of the total turning number below and above). Though the singular contour can be more complicated (e.g., connecting more handles), it will effectively convert the left-side picture into the right-side one: it would cause an additional cycle passing through the first singularity (like the cycle between the two singularities shown on the right), thus making the genus $g$ even.

I think this argument would generalize to a surface with more handles, as soon as any cycle exists between the "bottom" and "top" of the singular level of the type shown in the figure (left): in the absence of centers, for each saddle "that changes the orientation," along the evolution of the level sets there will occur another similar saddle adding a second cycle.

Unfortunately, I lack the skill to convert this into a formal proof, and even if I could do it for this particular type of immersion of $\mathbb RP^2$ (Boy surface), it would not prove the claim in the general case. Could you provide such a proof, or point to sources where a proof can be found? Detailed explanations would be greatly appreciated, since I am not an expert.