What is the actual space available? Can you provide us a drawing or picture with everything in it (RF module, main PCB, battery, housing and wiring) and the available space for the antenna?
Regarding measurement of impedance, yes you can with a slotted line or a home made microstrip. By first probing the standing wave pattern along the slotted line/microstrip you know the cable loss and the position of the reference plane. With the DUT connected, you can determine the actual reflection coefficient based on the standing wave pattern. It is elaborate, but it can give good result. It is one of my favourites during courses.
If you are only interested in VSWR, you could use a directional coupler, you may check Ebay or other trade site for a coupler for your frequency range. I sometimes use a 2…100 MHz coupler from mini circuits. If you don't have a spectrum analyzer or 400 MHz capable oscilloscope, you could use a diode probe (power probe). Other option is a downconverter with LO and viewing the IF frequency on an oscilloscope. another option is to build a wheatstone bridge with floating diode detector as shown here:
Homebrew SWR meter for VHF, UHF and SHF (including Wifi ISM Band). There are many varieties. When using SMD and good RF construction, performance can be good well over 2 GHz.
As far as I know, most HAM (licensed radio amateur) VNA's do not include 433 MHz, but you may give it a try
A complete other issue (even when having the best VNA) is that when you connect the feed cable to your circuit, the shield of the cable becomes more or less part of the antenna. Especially with circuit with small ground planes, this will severally affect the measurement when using monopole antennas. The impedance is not the value for the monopole alone, but a combined (not summed) value for both the (small) ground plane and the monopole.
To avoid the cable coupling issue, you need common mode decoupling. This can be done with ferrites (yes also at 433 MHz) and/or resonating/quarter wave traps or other means. With very small ground planes and quarter wave whips, it is hardly possible to get sufficient decoupling. I then us the whip itself as ground for the feed cable.
Regarding the design itself, if you don't have any ground/counterpoise provision, you may consider a tuned loop antenna. In case of a hand held device (for example a remote control), you may use the hand as capacitive groundplane.