Thank you Duda,
First, if Luka is not dehydrated (both by his vet's lack of diagnosis of this and your not seeing those signs I noted), then IV fluids would not necessarily be ideal here. We would want to keep a close eye on his hydration at this stage, but the reason we'd not rush to use IV fluids in a non-dehydrated cat has to do with his borderline anemia. Because, as I am sure you can appreciate, administering fluids to a cat who is not dehydrated and has (near) anemia, could actually run the risk of diluting his blood and worsening the anemia situation. Furthermore, if a cat is not dehydrated, then fluid therapy would not necessarily be indicated since one could overhydrate a cat in this manner which could lead to complications with the heart and lungs (where fluid is lost from circulation into the lungs leading to breathing difficulties).
That said, if Luka is not drinking well and you wanted to try and support his hydration in a less invasive way, I would advise speaking to your vet about intermittent subcutaneous fluid administration. This is where we put sterile fluids under the skin for slow absorption. It is something the vet can usually do during a visit in the consult room with you (so Luka would not need to stay at the vets). As well, if Luka might need long term fluid support, then you could ask them to teach you how to give fluids subcutaneously at home (More Info). So, in this situation, if he is not dehydrated just now, then IV fluids with hospitalization may not be necessary and instead you can consider intermittent visits together to your vet or have them teach you how to help keep him hydrated in this less invasive manner.
Now in regards XXXXX XXXXX nutrition (also known as TPN or Total Parenteral Nutrition), this is not commonly administered to animals in general practice. When used this tends to be given at specialist facilities (ie vet schools -like the RVC). The reason is because as I am sure you can appreciate, this is a very rich preparation therefore any lapse in sterile technique during administration can lead to bacterial contamination and therefore a risk of bacteria getting into the patient's blood stream. Therefore, TPN is not used lightly and when administered would only be done in a specialist intensive care facility to prevent complications with administration. So, if he is not dehydrated (since this too could dilute his blood) and not in a severe state requiring referal to a vet school, the use of TPN is unlikely to be indicated here at this stage for Luka.
Now you noted that he has had an injection to address vomiting, which I am glad to see. Still, if his lack of appetite remains, you may need to speak to his vet about other anti-nausea options. Without knowing what type of injection this was, it is difficult for me to advise on alternatives but we do sometimes find that some cats will respond to some treatments better then others. Therefore, it may be a case of finding what works best for Luka.
Now if Luka isn't keen on the A/D, then you may want to consider using Royal Canin Recovery (LINK), or Clinicare Canine/Feline Liquid Diet (LINK). All of these are critical care diets that are calorically dense, so a little goes a long way nutrition-wise and we can get more calories in even if we can’t get a huge volume of food in. That said, if Luka is not keen to eat these on his own, you will need to consider potentially introducing syringe feeding with these feeds. With the A/D, all you need to do is add water to make this into a gruel for syringe feeding. It won't be a fun job, but as long as Luka is not actually vomiting, you can administer food this way.
Further to this, and less risk than TPN, you could speak to his vet about the potential for a feeding tube. Often when we have poorly kitties that we are trying to nurse through treatable conditions, we may find the use of feeding tubes useful as a short term means of keeping the nutrition into them even if they are not eating. You can read a good wee article on feeding tubes types and read about a few cases of their use HERE. So, this would be a consideration for Luka (and once the tube is in, you would be able to feed him through the tube at home).
Finally, if you try the above and he is still resistant to eating for you, then do consider speaking with his vet about an appetite stimulant drug (ie Cyproheptadine, Mirtazapine/ Mertazipine, etc) to encourage him to eat for you. Sometimes using a chemical "push" can remind them to eat for us and help get them eating again.
Overall, if Luka is not dehydrated, then hospitalisation for IV fluids would not necessarily be indicated for him. Still if he is not drinking well, then there is no reason why you cannot have subcutaneous fluids administered at his regular vets with your present. As well, if he may need fluids long term, they can teach you how to do this at home. Otherwise, IV nutrition would not be ideal at this stage and instead we'd first want to see if his vet can try an alternative anti-vomiting medication +/- appetite stimulating drugs to tempt Luka to eat for you. And if these do not break his fast, then we'd want to consider syringe feeding and/or a feeding tube to help get food into him to support him until he is ready to eat on his own again.
Finally, in case you did want to find another vet who can help you with this (or if you wanted an ER vet tonight), then you can use your post code to check the RCVS Register (HERE) to find your local Vets Now (LINK). Furthermore, if you wanted Luka to see a specialist internal medicine veterinary surgeon then I would suggest that the RVC (link above) is your closest veterinary school that your vet can refer Luka's care to.
I hope this information is helpful.
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All the best,
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