There is a way forward that will prove beneficial to a time that a New Cart' is introduced to the system, it will come with a cost, but due to the period of time owned devices have been around for, it will make sense to consider a service and maybe extend this to a change of some of the components.
There are no Warranty Concerns with the owned devices, if a odd change is made to a components as a result of a circuit inspection identifying the need.
Your devices at present in use for Amplification are Valve>Valve>SS.
I am not 100% sure but I would assume the PH3 has a JFET Circuit for a LOMC.
The MM Stage is most likely all Valve?
Your Old Style Sumiko SHO ' if with the Bulbous Wood Body ' will be able to be played directly through the MM Stage.
Are you familiar with using the MC Stage of the PH3?
I will list a few items that you might like to see an alternative approach and worthwhile considering.
1, If the Cart' is 'liked' and at the end of its usage life, it could be sent into VAS NY Inc for a assessment and refurb'. In general a Cart' can be dealt with (thorough clean and retip ) for a very fair sum. As an additional the Cart' can have more desirable materials added if compatible to improve on the mechanical interface. The Work undertaken by VAS NY Inc can seen in threads on Audiogon and can be witnessed on their Facebook Page.
2, The Valve Phono Pre' - As a result of the age it would not do any harm to have a the Circuit looked at and have measurements of the components taken. It might be time to swap out a part or two. If on the Signal Path, have a think toward very closely matched parts or even a Boutique Component such as a Z Foil or a Copper Cap'. There are 3 x 6922's in this design.
Have the 6922's tested for their usage life, this would be the first endeavour. If too inconvenient to arrange, going to the lengths of having a Matched Replacement Set Supplied, will give a reassurance that all is fine with the Valve.
A SUT or Head Amp might also need to be considered if a LOMC is the end choice, there is the possibility there is a JFET MC Stage on the Phono-Pre', and these from my experiences are not as attractive in use as a SUT or Head Amp.
Valves are great when it comes to trying them out, as the exchanger only has to be careful, the exchanges can be done without too much skill levels. It is worthwhile reading up on the impact of 6922 or their derivatives to see what ones might catch your curiosity, with 3 x Tubes this is relatively affordable.
3, The Pre-Amp' - This is pretty much the same as for the Phono Pre', but when it comes to the Valves, there are 8 x 6922's and 2 x 5881's.
The 5881's are looking like the ones to work with as the initial due to the quantity and affordability of a Matched Pair of these or a derivative.
I have never Tube Rolled any Item with such a quantity of the same Valve as the Pre's 6922's. Getting matched Valves as an Octet Set are a new one to me and will probably be a few hundred $'s for New Models and move quickly in ascending value when NOS are considered. Either way the ones in use will benefit from a Test to show the usage life.
I have thrown a Curve Ball, but I don't know any other way to approach the use of aged Valve Equipment where I have no knowledge of the maintenance or usage.
I have a Valve Power Amp' built for myself in the early 90's, that has been used intermittently since the early 00's, usually for a demonstration, even though I have no confidence it is a optimised working Amp.
My wife is requesting that I create a Small System that she can replay her Vinyl Albums at her leisure, without navigating the main HiFi System. I have bought Matched Input Tubes and Matched K5881 Output Tubes for the Amp in advance of sending it to a EE to check out the circuits, this one will most likely be added to the system requested by the wife.
My approach to your inquiry, is born from the knowledge of which Cart' Types are compatible with your SME V. Your own supplied list of Brands is going to catapult you into the $2K -$5K suggestions.
Using a Cart' of even a $1000+ in conjunction with a downstream electronics that is aged and not optimised as a Circuit will create a diminished performance from the Cart' chosen.
I am suggesting getting the best from what you already have prior to committing to what looks like, will eventually be a LOMC.